A unified thermodynamic picture of Horava-Lifshitz black hole in arbitrary space time

Abstract

Abstract

In this paper, we analyze the complete thermodynamic and phase transition phenomena of a black hole solution in Horava-Lifshitz gravity in arbitrary space time. Nature of phase transition is studied using geometrothermodynamic and Ehrenfest's scheme of standard thermodynamics. We analytically check the Ehrenfest's equations near the critical point, which is the point of divergence in the heat capacity. Our analysis revels that this black hole exhibits a second order phase transition.

Published for SISSA by Springer Received: August 6, 2014 Revised: November 30, 2014 Accepted: December 15, 2014

Published: January 8, 2015

JHEP01(2015)019

A unied thermodynamic picture of Hoava-Lifshitz black hole in arbitrary space time

Jishnu Suresh, R. Tharanath and V.C. KuriakoseDepartment of Physics, Cochin University of Science and Technology, Kochi, 682 022, Kerala, India

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: In this paper, we analyze the complete thermodynamic and phase transition phenomena of a black hole solution in Hoava-Lifshitz gravity in arbitrary space time. Nature of phase transition is studied using geometrothermodynamic and Ehrenfests scheme of standard thermodynamics. We analytically check the Ehrenfests equations near the critical point, which is the point of divergence in the heat capacity. Our analysis revels that this black hole exhibits a second order phase transition.

Keywords: Models of Quantum Gravity, Black Holes, Classical Theories of Gravity

ArXiv ePrint: 1408.0911

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP01(2015)019

Web End =10.1007/JHEP01(2015)019

Contents

1 Introduction 1

2 Review of thermodynamics of LMP black hole 2

3 Geometrothermodynamics 8

4 Analytical check of classical Ehrenfest equations 9

5 Conclusion 12

1 Introduction

Black hole thermodynamics has been a fascinating topic of study since black holes were identied as thermodynamic objects with both temperature and entropy [1, 2]. Thermodynamic properties of black holes have been studied during all these years. During this period, it became well known that the black hole spacetime can possess phase structures along with the standard thermodynamic variables like temperature, entropy etc. Hence it causes to believe in the existence of a complete analogy between black hole system and non-gravitational thermodynamic systems. Various black hole thermodynamic variables and their properties have been extensively studied. In 1983, Hawking and Page [3] discovered the phase transition phenomena in the Schwarzschild AdS background. This became a turning point in the study of black hole phase transition. After this, many studies have been done in this regard [6287].

Recently geometric method has been identied as a convenient tool to study the thermodynamics and the corresponding phase transition structures of black holes. Various investigations are also done by incorporating this idea from information geometry to the study of black hole thermodynamics [48]. Riemannian geometry to the equilibrium space was rst introduced by Weinhold and Ruppeiner. In 1976 Weinhold [913] proposed a metric, as the Hessian of the internal energy, given as gWij = @i@jU (S, Nr). Later in 1979, Ruppeiner [14] introduced another metric as the Hessian of entropy, as gRij = @i@jS (M, Nr).

This Ruppeiner metric is conformally equivalent to Weinholds metric and the geometry that can be obtained from these two methods are related through the relation where [15, 16], ds2R =

1

T ds2W . Since these matrices depend on the choice of thermodynamic potentials, they are not Legendre invariant. The results obtained with the above two metrics are found to be consistent with the systems like ideal classical gas, multicomponent ideal gas, ideal quantum gas, one-dimensional Ising model, van der Waals model etc, [1727]. But these two metrics fail in explaining the thermodynamic properties and they lead to many puzzling situations. By incorporating the idea of Legendre invariance, Quevedo et al. [4850]

1

JHEP01(2015)019

proposed a new geometric formalism, known as the Geometrothermodynamics. The metric structure related with geometrothermodynamics can give well explanations for di erent behaviors of black hole thermodynamic variables. This method seizes the exact phase structure of black hole systems.

On the other hand, classical thermodynamics can be applied to black hole systems directly to study the phase structure. In classical thermodynamics rst order phase transitions satisfy Clausius-Clapeyron equation, while second order phase transitions satisfy Ehrenfest equations. Recently Banerjee et al. [5358] developed a new scheme to study the phase transition in black holes based on Ehrenfest equations by considering the analogy between thermodynamic variables and black hole parameters. This Ehrenfest scheme provides a unique way to classify the nature of phase transitions in black hole systems. If two Ehrenfest relations are satised by a black hole system, then the corresponding phase transition can be identied as second order in nature. Even if it is not second order, we can nd the deviation from second order by dening Ehrenfests relation as quantied by dening a parameter called the Prigogine-Defay(PD) ratio [5961]. Interestingly many calculations show that there are black hole systems whose phase transitions lie within the bound conforming to a glassy phase transition.

Hoava-Lifshitz theory is an anisotropic and a non-relativistic renormalizable theory of gravity at a Lifshitz point, which can be treated as a good candidate for the study of quantum eld theory of gravity, and it retrieves the Einsteins gravity in the IR limit. Recently its black hole solution and thermodynamics have been intensively investigated [28, 3336, 3847]. By introducing a dynamical parameter in asymptotically AdS4 space-time, a spherically symmetric black hole solution was rst given by Lu et al. [28]. In this paper, motivated by all the above mentioned features, we extract the whole thermodynamic quantities of this black hole, called Lu Mei Pope (LMP) black hole in arbitrary space time. Also we study the phase structure of this solution using geometric methods as well as using the Ehrenfest scheme.

The paper is organized as follows. In section 2, we discuss the thermodynamics of LMP black hole in HL gravity and we calculate the equation of state of the black hole system. Using the idea of geometrothermodynamics, the thermodynamics and the peculiar behaviors of thermodynamic variables are studied in section 3. In section 4, using the Ehrenfest scheme, the nature of phase transition is discussed. Finally the conclusions are quoted in section 5.

2 Review of thermodynamics of LMP black hole

Hoava used the ADM formalism, where the four-dimensional metric of general relativity is parameterised as [29],

ds24 = N2dt2 + gij(dxi Nidt)(dxj Njdt), (2.1)

2

JHEP01(2015)019

where N, Ni and gij are the lapse, shift and 3-metric respectively. The ADM decomposition (2.1) of the Einstein-Hilbert action is given by,

SEH = 116G

[integraldisplay]

d4xpgN(KijKij K2 + R 2 ), (2.2)

where G is Newtons constant, R is the curvature scalar and Kij is dened by,

Kij = 12N ( gij riNj rjNi). (2.3)

The action of the theory proposed by Hoava [30] can be written as,

SHL =

[integraldisplay]

dtd3xpgN[

2 2 (KijKij K2) +

2[notdef]2( W R 3 2W )

8(1 3 )

Cij

[notdef]w22 Rij

[parenrightbigg] [parenleftbigg]

Cij

[notdef]w22 Rij[parenrightbigg]], (2.4)

where , , [notdef] , w and W are constant parameters, and Cij is the Cotton tensor, dened by,

Cij = [epsilon1]ik[lscript]rk [parenleftbigg]

+

2[notdef]2(1 4 )32(1 3 )

R2

2 2w4

1 4R j[lscript][parenrightbigg]

. (2.5)

Comparing the rst term in (2.4) with that of general relativity in the ADM formalism, we can write the speed of light, Newtons constant and the cosmological constant respectively as,

c = 2[notdef] 4

[radicalbigg]

W1 3

JHEP01(2015)019

Rj[lscript]

, G = 2

32 W . (2.6)

Now we will look for a static and spherically symmetric solution with the metric,

ds2 = f(r)dt2

dr2f(r) + r2d 2. (2.7)

In the present study we are interested in the solution with the choice = 1. This will lead to the LMP black hole solution [28], given by,

f(r) = k W r2 A[radicalbigg]

32c and =

r

W , (2.8)

where A is an integration constant and is related to the black hole mass as A = aM. It is

interesting to note that this solution (2.8) is asymptotically AdS4. From the rst law of black hole mechanics [31], when a black hole undergoes a change from a stationary state to another, then the change in mass of the black hole is given by,

dM =

8 dA + dJ + dQ. (2.9)

Comparing this with the rst law of thermodynamics,

dM = T dS + dJ + dQ, (2.10)

3

f [LParen1] r [RParen1]

15

10

5

JHEP01(2015)019

1 2 3 4

r

Figure 1. Plots of f(r) vs r for k = 0 (solid line), k = 1 (dotted line) and k = 1 (dashed line)

with W = 1, a = 1, M = 1.

Figure 2. 3D Plots of Mass vs rh.

one can easily establish the analogy between black hole mechanics and the rst law of thermodynamics . We know that Hoava-Lifshitz theory does not possess the full di eo-morphism invariance of general relativity but only a subset in the form of local Galilean invariance. This subset is manifest in the Arnowitt, Deser and Misner (ADM) slicing. Here we have considered the ADM decomposition of the four dimensional metric. Then for a non-rotating uncharged black hole, the entropy can be written as [36, 37],

S = [integraldisplay]

T =

[integraldisplay]

@H

@rh drh, (2.11)

where H denotes the enthalpy and rh denotes the horizon radius. And the Hawking temperature can be determined from,

TH = 2 =

14 f[prime](r)[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

r=rh

, (2.12)

dM

1 TH

4

JHEP01(2015)019

Figure 3. 3D Plots of temperature.

TH =

S

. (2.15)

The heat capacity at constant pressure and at constant volume can be obtained respectively as,

CP = T

and,

W . (2.18)

The event horizon can be obtained from f(rh) = 0, and from that one can easily arrive at the black hole mass-event horizon radius relation as,

M = 1

a

5

and,

@H @S

P

, (2.13)

where P is the pressure and it is related to the Hawking temperature and entropy as,

P = 12THS. (2.14) Hence the volume of the black hole is given by,

V =

@H @P

@S @T

P

, (2.16)

CV = CP + V @P

@T . (2.17) Using these relations we can calculate the thermodynamic quantities of the LMP black holes in arbitrary space curvature. In the cases of spherical(k = 1) and at spaces (k = 0), detailed studies are done in [32]. And it is noted that, in both cases the black hole doesnt show any kind of phase transition behaviors. In this paper we are interested in the LMP black hole solution in Hyperbolic space (k = 1). In this case, (2.8) can be reduced to,

f(r) = 1 W r2 A[radicalbigg]

r

r W

rh 1 W r2h

[parenrightbig]

. (2.19)

In gure 2 we draw the 3D plot of variation of black hole mass with respect to black hole horizon radius for varying cosmological constant term( W ). From this gure it can be easily seen that the black hole mass increases with increase in the magnitude of the cosmological constant. It is interesting to note from this gure that the black hole vanishes for lower values of S. Considering the numerical values, say a = 1 and W = 1 the black hole vanishes at rh = 1, but when W = 2 the black hole vanishes at another value of

horizon radius, rh = 0.5, and so on.Black hole entropy can be obtained from (2.11) as,

S = 8p W

a . (2.20)

From (2.12) we can derive the Hawking temperature as,

TH = 1 3 W r2h 8rh , (2.21)

3D plot of Hawking temperature with respect to black hole horizon radius for varying cosmological constant term is depicted in gure 3. Here also the temperature increases with the magnitude of the cosmological constant. From (2.14), black hole pressure can be found as

P = 21rh 1 3 W r2h 64 . (2.22)

From the above expression, it is obvious that for any negative value of the cosmological constant term W , the pressure is found to be positive. Using (2.15), black hole volume is given by,

V = 32p W21ar

3 2

h

JHEP01(2015)019

1 3 W r2h

. (2.23)

Using (2.16) and (2.17), the heat capacity at constant pressure and at constant volume are respectively determined as,

CP = 4p

1 9 W r2h

W rh a

3 W r2h 1

[parenrightbig]

, (2.24)

3 W r2h + 1

[parenrightbig]

and,

CV = 8p

W rh a

3 W r2h 1

[parenrightbig]

. (2.25)

In gure 4 we draw the 3D variations of heat capacity with respect to horizon radius and for varying cosmological constant term. From this gure, it is evident that the black hole has both positive and negative values in certain parametric regions. It is also clear from the gure that, heat capacity has a divergent point. According to Davies [88], second order phase transitions takes place at those points where the heat capacity diverges. So LMP black hole undergoes a phase transition in this case. By investigating the free energy of the black hole we can get a clear picture of the phase transition. Free energy of the black hole is given by,

F = M T S. (2.26)

6

3 W r2h + 1

[parenrightbig]

JHEP01(2015)019

Figure 4. 3D Plots of heat capacity.

0.8

T

0.7

0.6

0.5

0.4

0.3

[Minus] 20 [Minus] 15 [Minus] 10 [Minus] 5

F

Figure 5. Parametric plot of free energy and temperature for W = 1, a = 1.

Using (2.19), (2.20) and (2.21) free energy of LMP black hole is obtained as,

F = 1

a

r W rh

2(1 + W r2h)(1 9 W r2h) + (3 W r2h 1)22(9 W r2h 1) [parenrightbigg]

. (2.27)

Now one can plot the parametric variation of free energy and temperature using (2.27) and (2.21). From gure 5, it is evident that there is a cusp like double point. This indicates a second order phase transition. For one of the branches the free energy decreases and reaches the temperature which corresponds to the minimum free energy. There after, free energy increases with a di erent slope.

An equation of state in general, is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy and black hole equation of state can be written from (2.22), (2.23) and (2.21) as,

P V

43 = 4T

3

r3263 W

a2 . (2.28)

7

Figure 6. Isotherm P V diagram.

Here, P is the the pressure, V is the thermodynamic volume and T is the black hole temperature. Now, we plot the isotherm P V diagram in gure 6. From (2.28) and

the corresponding gure (gure 6) we can conclude that the behaviour is ideal gas like. Hence no critical point can be found and there would be no P V criticality.

3 Geometrothermodynamics

In order to introduce the idea of di erential geometry to thermodynamics, according to [48 50], we have to dene a (2n + 1) dimensional thermodynamic phase space T . It can be

coordinated by the set ZA = [notdef] , Ea, Ia[notdef], where represents the thermodynamic potential

and Ea and Ia represent extensive and intensive thermodynamic variables respectively. The phase space is provided with the Gibbs 1-form = d abIadEb, satisfying the

condition, ^ (d )n [negationslash]= 0. Consider a Riemannian metric G on T , which must be invariant

with respect to Legendre transformations. Then the Riemannian contact manifold can be dened as the set (T , , G), and the equilibrium manifold can be written as a sub

manifold of T , i.e., E T . This sub manifold satises the condition ' ( ) = 0 (where

' : E ! T ), known as the pull back condition [61]. Then the non-degenerate metric G and

the thermodynamic metric g can be written as,

G = (d abIadEb)2 + ( abEaIb)( cddEcdId), (3.1)

and,

gQ = ' (G) =

Ec @

@Ec

[parenrightbigg] [parenleftbigg]

ab bc @2

@Ec@Ed dEadEd[parenrightbigg]

, (3.2)

with ab=diag(-1,1,1,. . . ,1) and this metric is Legendre invariant because of the invariance of the Gibbs 1-form.

Now we will introduce this idea of Geometrothermodynamics in to the LMP black hole system to study whether the black hole exhibits a phase transition or not. For this, we will consider a 5-dimensional thermodynamic phase space T constituted by extensive variables

8

JHEP01(2015)019

R

0.8

0.6

0.4

0.2

[Minus] 0.2

[Minus] 0.4

S

1 2 3 4 5

Figure 7. Variation of scalar curvature with horizon radius for W = 1, a = 1.

S and a and the corresponding intensive variables T and A. Thus the fundamental 1-form dened on T can be written as,d = dM T dS a dA . (3.3)

Now the Quevedo metric [48] is given by,

g = (SMS + aMa)

[bracketleftBigg]

Then the Legendre invariant scalar curvature corresponding to the above metric is given by,

R = 48a2r3 35r6 3W 5r4 2W + 9r2 W + 9

(r2 W + 3) 2 (3r2 W + 1) 2 (5r2 W 3) 3. (3.4)

We have plotted the variation of scalar curvature with horizon radius in gure 7. From this gure as well as from the above equation, it can be conrmed that this scalar curvature diverges at the same point where the heat capacity diverges. Hence the Geometrothermo-dynamics exactly reproduces the phase transition structure of the LMP black hole.

4 Analytical check of classical Ehrenfest equations

Innite discontinuity in the heat capacity of the black hole does not always indicate a second order phase transition, but it suggests the possibility of a higher order phase transition. In classical thermodynamics, one can conrm the rst order phase transition by utilizing Clausius-Clapeyron equations. Similarly a second order transition can be conrmed by checking whether it satises Ehrenfest equations or not. The original expressions of Ehrenfest equations in classical thermodynamics are given by,

@P @T S= CP2 CP1

V T ( 2 1)= CP

V T , (4.1)

@P @T V= 2 1 T2 T1= , (4.2)

9

JHEP01(2015)019

MSS 0

0 Maa

[bracketrightBigg]

.

where = 1

V

@V

@T

P is the volume expansion coe cient and T = 1V

@V @P

T is the isothermal compressibility coe cient. Considering the analogy between the thermodynamic variables and black hole parameters, where pressure (P ) is replaced by the negative of the electrostatic potential di erence ( ), and volume (V ) is replaced by charge of the black

hole (Q). Thus for black hole thermodynamics, the two Ehrenfest equations (4.1) and (4.2) become,

@ @T

S= 1QT

C 2 C 1( 2 1)= C

QT , (4.3)

@ @T

Q= 2 1 T2 T1= , (4.4)

where = 1

Q

is the volume expansion coe cient and T = 1Q [parenleftBig] @Q @ T is the isothermal compressibility coe cient of the black hole system. Here, in the above sets of equations, the subscripts 1 and 2 denote two distinct phases of the system.

In this paper, rather than considering the black hole analogy of Ehrenfest equation, we will introduce the classical Ehrenfest equation directly in to the black hole system under consideration. Using (2.20), (2.21), (2.22) and (2.23), we can arrive at the expressions of specic heat at constant pressure, volume expansion coe cient and isothermal compressibility coe cient respectively as,

CP = 4p

JHEP01(2015)019

@Q @T

W rh a

3 W r2h 1

3 W r2h + 1

, (4.5)

= 6rh

1 3r2h W

, (4.6)

and,

= 16

7rh

11 3r2h W

. (4.7)

From these relations, it is interesting to note that both volume expansion coe cient and isothermal compressibility coe cient have same factor in the denominator, which implies that both these parameters diverge at the same point. We have plotted the variation of these coe cients with respect to the horizon radius (rh) in gures 8 and 9 respectively.

Now we will investigate the nature of phase transition at the critical point of LMP black hole by doing the analytic check of classical Ehrenfest equations (4.1) and (4.2). The values of temperature, pressure and volume at the critical point are respectively given by,

Tc = p3p

W

4 , (4.8)

Pc = 7p3

32p W , (4.9)

and,

Vc = 16( W )5[notdef]4

7 4p3a . (4.10)

10

5

a

4

3

2

1

JHEP01(2015)019

2 4 6 8 10 12 14

rh

Figure 8. Variation of volume expansion coe cient with horizon radius for W = 1, a = 1.

0.30

k

0.25

0.20

0.15

0.10

0.05

2 4 6 8 10

rh

Figure 9. Variation of isothermal compressibility with horizon radius for W = 1, a = 1.

Now lets check the validity of Ehrenfest equations at the critical point. From the denition of volume expansion coe cient , given by (4.6), we obtain,

V =

CP

T

[parenrightbigg]

, (4.11)

then, the r.h.s. of rst classical Ehrenfest equation (4.1) becomes,

CP

T V =

[bracketleftbigg][parenleftbigg]

@V @T

P=

@V @S

@S @T

P=

@V @S

P

P

@S @V

P

rcri, (4.12)

where rcri denotes the critical point. Applying the above equation to LMP black hole system, we obtain,

CP

T V =

21 24

1

W . (4.13)

Now the l.h.s. of rst classical Ehrenfest equation (4.1) becomes,

@P @T

S

rcri= 21241

W . (4.14)

11

From (4.14) and (4.13), we can arrive at the conclusion that both l.h.s. and r.h.s. of rst Ehrenfest equation are in good agreement at the critical point rcri. From (4.6) and (4.7), using the thermodynamic identity,

@V @P T

@P @T

@T @V

P= 1, (4.15)

we can obtain,

V

V T =

@V @P

T=

@T @P

@V @T

P=

@T @P

V

V . (4.16)

V

JHEP01(2015)019

Now the r.h.s. of (4.2) can be obtained as,

T = [bracketleftbigg][parenleftbigg]

@P

@T

V

rcri= 21241( W )43. (4.17)

Also the l.h.s. of (4.2) can be obtained as,

[bracketleftbigg][parenleftbigg]

@P

@T

V

rcri= 21241( W )43. (4.18)

From (4.18) and (4.17), we can conclude that second Ehrenfest equation is satised at the critical points. Hence both the Ehrenfest equations are in good agreement at the critical point. Using (4.13) and (4.17), the Prigogine-Defay (PD) can be obtained as

= CP T

T V ( )2 = 1. (4.19)

This conrms that the phase transition of LMP black hole in Hoava-Lifshitz gravity is second order in nature.

5 Conclusion

The complete thermodynamics and phase transition picture of LMP black holes in Hoava-Lifshitz gravity have been investigated using both thermodynamic geometry and Ehrenfests scheme. We have systematically analyzed the thermodynamics and phase transition. From this thermodynamic study, absence of any discontinuity in entropy-temperature relationship eliminates the presence of any rst order transition. Then the heat capacity is found to be diverging, thereby indicating the presence of a phase transition. But the order of phase transition was not revealed. To further clarify the existence of phase transition, geometrothermodynamics is applied, in which the critical point where the heat capacity diverges coincides with the diverging point of Legendre invariant geometrothermodynamic scalar curvature. Hence GTD metric exactly reproduces the phase transition structure of LMP black hole and their corresponding thermodynamic interactions. Here we can conclude that the curvature scalar behaves in a similar way as that of the black hole system. Then we have conducted a detailed analytic check of classical Ehrenfest equations on LMP black hole system. From this we have found that the LMP black hole satises the Ehrenfest

12

equations, and hence the phase transition is second order in nature. The PD ratio found in these calculations further witnesses the second order phase transition, and conrms that there is no deviation from this order. Hence it will be possible to answer whether one can have a quantum eld theory at a nite temperature by studying the thermodynamic stability of the black hole, as evident from the specic heat. However, the black hole conguration must be favourable over pure thermal radiation in anti-de Sitter space, that is, have dominant negative free energy. The present black hole solution satises these conditions. Hence from this study one can look forward for the implication on the dual eld theory which exists on the boundary of the anti-de Sitter space.

Acknowledgments

The authors wish to thank referee for the useful comments and UGC, New Delhi for nancial support through a major research project sanctioned to VCK. VCK also wishes to acknowledge Associateship of IUCAA, Pune, India.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] J.D. Bekenstein, Black holes and entropy, http://dx.doi.org/10.1103/PhysRevD.7.2333

Web End =Phys. Rev. D 7 (1973) 2333 [http://inspirehep.net/search?p=find+J+Phys.Rev.,D7,2333

Web End =INSPIRE ].[2] S.W. Hawking, Particle creation by black holes, http://dx.doi.org/10.1007/BF02345020

Web End =Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [http://inspirehep.net/search?p=find+J+Comm.Math.Phys.,43,199

Web End =INSPIRE ].

[3] S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, http://dx.doi.org/10.1007/BF01208266

Web End =Commun. Math. Phys. 87 (1983) 577 [http://inspirehep.net/search?p=find+J+Comm.Math.Phys.,87,577

Web End =INSPIRE ].

[4] J. Gibbs., The collected works. Volume 1: thermodynamics, Yale University Press, New Haven, U.S.A. (1948).

[5] C. Caratheodory, Untersuchungenber die Grundlagen der Thermodynamik, Gesammelte Mathematische Werke Band 2, Munich, Germany (1995).

[6] R. Hermann, Geometry physics and systems, Marcel Dekker, New York U.S.A. (1973).[7] R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics, Rep. Math. Phys. 14 (1978) 419.

[8] R. Mrugala, Submanifolds in the thermodynamical phase space, Rep. Math. Phys. 21 (1985) 197.

[9] F. Weinhold, Metric geometry of equilibrium thermodynamics I, J. Chem. Phys. 63 (1975) 2479.

[10] F. Weinhold, Metric geometry of equilibrium thermodynamics II, J. Chem. Phys. 63 (1975) 2484.

[11] F. Weinhold, Metric geometry of equilibrium thermodynamics III, J. Chem. Phys. 63 (1975) 2488.

[12] F. Weinhold, Metric geometry of equilibrium thermodynamics IV, J. Chem. Phys. 63 (1975) 2496.

13

JHEP01(2015)019

[13] F. Weinhold, Metric geometry of equilibrium thermodynamics V, J. Chem. Phys. 65 (1976) 558.

[14] G. Ruppeiner, Thermodynamics: a Riemannian geometric model, http://dx.doi.org/10.1103/PhysRevA.20.1608

Web End =Phys. Rev. A 20 (1979) 1608.

[15] R. Mrugala, On equivalence of two metrics in classical thermodynamics, http://dx.doi.org/10.1016/0378-4371(84)90074-8

Web End =Physica A 125 http://dx.doi.org/10.1016/0378-4371(84)90074-8

Web End =(1984) 631 .

[16] P. Salamon, J.D. Nulton and E. Ihrig, On the relation between entropy and energy versions of thermodynamic length, J. Chem. Phys. 80 (1984) 436.

[17] H. Janyszek, On the geometrical structure of the generalized quantum Gibbs states, http://dx.doi.org/10.1016/0034-4877(86)90037-6

Web End =Rep. http://dx.doi.org/10.1016/0034-4877(86)90037-6

Web End =Math. Phys. 24 (1986) 1 .

[18] H. Janyszek and R. Mrugala, Riemannian geometry and the thermodynamics of model magnetic systems, http://dx.doi.org/10.1103/PhysRevA.39.6515

Web End =Phys. Rev. A 39 (1989) 6515 [http://inspirehep.net/search?p=find+J+Phys.Rev.,A39,6515

Web End =INSPIRE ].

[19] E.J. Brody, Applications of the Kakutani metric to real space renormalization, http://dx.doi.org/10.1103/PhysRevLett.58.179

Web End =Phys. Rev. http://dx.doi.org/10.1103/PhysRevLett.58.179

Web End =Lett. 58 (1987) 179 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,58,179

Web End =INSPIRE ].

[20] D. Brody and N. Rivier, Geometrical aspects of statistical mechanics , http://dx.doi.org/10.1103/PhysRevE.51.1006

Web End =Phys. Rev. E 51 http://dx.doi.org/10.1103/PhysRevE.51.1006

Web End =(1995) 1006 [http://inspirehep.net/search?p=find+J+PHRVA,E51,1006

Web End =INSPIRE ].

[21] D. Brody and A. Ritz, On the symmetry of real space renormalization, http://dx.doi.org/10.1016/S0550-3213(98)00298-3

Web End =Nucl. Phys. B 522 http://dx.doi.org/10.1016/S0550-3213(98)00298-3

Web End =(1998) 588 [http://arxiv.org/abs/hep-th/9709175

Web End =hep-th/9709175 ] [http://inspirehep.net/search?p=find+J+NUPHA,B522,588

Web End =INSPIRE ].

[22] B.P. Dolan, Geometry and thermodynamic uctuations of the Ising model on a Bethe lattice, Proc. Roy. Soc. London A 454 (1998) 2655 [http://arxiv.org/abs/cond-mat/9706238

Web End =cond-mat/9706238 ].

[23] B.P. Dolan, D.A. Johnston and R. Kenna, The information geometry of the one-dimensional Potts model, http://dx.doi.org/10.1088/0305-4470/35/43/303

Web End =J. Phys. A 35 (2002) 9025 [http://arxiv.org/abs/cond-mat/0207180

Web End =cond-mat/0207180 ] [http://inspirehep.net/search?p=find+J+J.Phys.,A35,9025

Web End =INSPIRE ].

[24] W. Janke, D.A. Johnston and R.P. K.C. Malmini, The information geometry of the Ising model on planar random graphs, http://dx.doi.org/10.1103/PhysRevE.66.056119

Web End =Phys. Rev. E 66 (2002) 056119 [http://arxiv.org/abs/cond-mat/0207573

Web End =cond-mat/0207573 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,E66,056119

Web End =INSPIRE ].

[25] W. Janke, D.A. Johnston and R. Kenna, The information geometry of the spherical model, http://dx.doi.org/10.1103/PhysRevE.67.046106

Web End =Phys. Rev. E 67 (2003) 046106 [http://arxiv.org/abs/cond-mat/0210571

Web End =cond-mat/0210571 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,E67,046106

Web End =INSPIRE ].

[26] W. Janke, D.A. Johnston and R. Kenna, Information geometry and phase transitions, Physica A 336 (2004) 181 [http://arxiv.org/abs/cond-mat/0401092

Web End =cond-mat/0401092 ] [http://inspirehep.net/search?p=find+J+Physica,A336,181

Web End =INSPIRE ].

[27] D.A. Johnston, W. Janke and R. Kenna, Information geometry, one, two, three (and four), Acta Phys. Polon. B 34 (2003) 4923 [http://arxiv.org/abs/cond-mat/0308316

Web End =cond-mat/0308316 ] [http://inspirehep.net/search?p=find+J+ActaPhys.Polon.,B34,4923

Web End =INSPIRE ].

[28] H. L, J. Mei and C.N. Pope, Solutions to Hoava gravity, http://dx.doi.org/10.1103/PhysRevLett.103.091301

Web End =Phys. Rev. Lett. 103 (2009) http://dx.doi.org/10.1103/PhysRevLett.103.091301

Web End =091301 [http://arxiv.org/abs/0904.1595

Web End =arXiv:0904.1595 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,103,091301

Web End =INSPIRE ].

[29] R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, http://dx.doi.org/10.1007/s10714-008-0661-1

Web End =Gen. Rel. http://dx.doi.org/10.1007/s10714-008-0661-1

Web End =Grav. 40 (2008) 1997 [http://arxiv.org/abs/gr-qc/0405109

Web End =gr-qc/0405109 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/0405109

Web End =INSPIRE ].

[30] P. Hoava, Quantum gravity at a Lifshitz point, http://dx.doi.org/10.1103/PhysRevD.79.084008

Web End =Phys. Rev. D 79 (2009) 084008 [http://arxiv.org/abs/0901.3775

Web End =arXiv:0901.3775 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D79,084008

Web End =INSPIRE ].

[31] J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, http://dx.doi.org/10.1007/BF01645742

Web End =Commun. Math. Phys. 31 (1973) 161 [http://inspirehep.net/search?p=find+J+Comm.Math.Phys.,31,161

Web End =INSPIRE ].

[32] J. Sadeghi, K. Jafarzade and B. Pourhassan, Thermodynamical quantities of Hoava-Lifshitz black hole, http://dx.doi.org/10.1007/s10773-012-1281-9

Web End =Int. J. Theor. Phys. 51 (2012) 3891 .

[33] J. Simon, Small black holes versus horizonless solutions in AdS, http://dx.doi.org/10.1103/PhysRevD.81.024003

Web End =Phys. Rev. D 81 (2010) http://dx.doi.org/10.1103/PhysRevD.81.024003

Web End =024003 [http://arxiv.org/abs/0910.3225

Web End =arXiv:0910.3225 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D81,024003

Web End =INSPIRE ].

14

JHEP01(2015)019

[34] A. Kehagias and K. Sfetsos, The black hole and FRW geometries of non-relativistic gravity, http://dx.doi.org/10.1016/j.physletb.2009.06.019

Web End =Phys. Lett. B 678 (2009) 123 [http://arxiv.org/abs/0905.0477

Web End =arXiv:0905.0477 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B678,123

Web End =INSPIRE ].

[35] H. Nastase, On IR solution in Hoava gravity theories, http://arxiv.org/abs/0904.3604

Web End =arXiv:0904.3604 [http://inspirehep.net/search?p=find+EPRINT+arXiv:0904.3604

Web End =INSPIRE ].[36] E. Kiritsis and G. Konas, Hoava-Lifshitz cosmology, http://dx.doi.org/10.1016/j.nuclphysb.2009.05.005

Web End =Nucl. Phys. B 821 (2009) 467 [http://arxiv.org/abs/0904.1334

Web End =arXiv:0904.1334 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B821,467

Web End =INSPIRE ].

[37] A.J. Padilla, The good, the bad and the ugly . . . of Hoava gravity, http://dx.doi.org/10.1088/1742-6596/259/1/012033

Web End =J. Phys. Conf. Ser. 259 http://dx.doi.org/10.1088/1742-6596/259/1/012033

Web End =(2010) 012033 [http://arxiv.org/abs/1009.4074

Web End =arXiv:1009.4074 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.4074

Web End =INSPIRE ].

[38] G. Calcagni, Cosmology of the Lifshitz universe, http://dx.doi.org/10.1088/1126-6708/2009/09/112

Web End =JHEP 09 (2009) 112 [http://arxiv.org/abs/0904.0829

Web End =arXiv:0904.0829 ] [http://inspirehep.net/search?p=find+J+JHEP,0909,112

Web End =INSPIRE ].

[39] M.-i. Park, A test of Hoava gravity: the dark energy, http://dx.doi.org/10.1088/1475-7516/2010/01/001

Web End =JCAP 01 (2010) 001 [http://arxiv.org/abs/0906.4275

Web End =arXiv:0906.4275 ] [http://inspirehep.net/search?p=find+J+JCAP,1001,001

Web End =INSPIRE ].

[40] S.W. Wei, Y.X. Liu and H. Guo, Thermodynamic geometry of black hole in the deformed Hoava-Lifshitz gravity, http://dx.doi.org/10.1209/0295-5075/99/20004

Web End =Europhys. Lett. 99 (2012) 20004 .

[41] Y.S. Myung, Entropy of black holes in the deformed Hoava-Lifshitz gravity, http://dx.doi.org/10.1016/j.physletb.2010.01.014

Web End =Phys. Lett. B http://dx.doi.org/10.1016/j.physletb.2010.01.014

Web End =684 (2010) 158 [http://arxiv.org/abs/0908.4132

Web End =arXiv:0908.4132 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B684,158

Web End =INSPIRE ].

[42] Y.S. Myung, Thermodynamics of black holes in the deformed Hoava-Lifshitz gravity, http://dx.doi.org/10.1016/j.physletb.2009.06.013

Web End =Phys. http://dx.doi.org/10.1016/j.physletb.2009.06.013

Web End =Lett. B 678 (2009) 127 [http://arxiv.org/abs/0905.0957

Web End =arXiv:0905.0957 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B678,127

Web End =INSPIRE ].

[43] Y.S. Myung and Y.-W. Kim, Thermodynamics of Hoava-Lifshitz black holes, http://dx.doi.org/10.1140/epjc/s10052-010-1319-1

Web End =Eur. Phys. J. http://dx.doi.org/10.1140/epjc/s10052-010-1319-1

Web End =C 68 (2010) 265 [http://arxiv.org/abs/0905.0179

Web End =arXiv:0905.0179 ] [http://inspirehep.net/search?p=find+J+Eur.Phys.J.,C68,265

Web End =INSPIRE ].

[44] N. Varghese and V.C. Kuriakose, Evolution of electromagnetic and Dirac perturbations around a black hole in Hoava gravity, http://dx.doi.org/10.1142/S0217732311035936

Web End =Mod. Phys. Lett. A 26 (2011) 1645 [http://arxiv.org/abs/1010.0549

Web End =arXiv:1010.0549 ] [http://inspirehep.net/search?p=find+J+Mod.Phys.Lett.,A26,1645

Web End =INSPIRE ].

[45] M.-i. Park, The black hole and cosmological solutions in IR modied Hoava gravity, http://dx.doi.org/10.1088/1126-6708/2009/09/123

Web End =JHEP http://dx.doi.org/10.1088/1126-6708/2009/09/123

Web End =09 (2009) 123 [http://arxiv.org/abs/0905.4480

Web End =arXiv:0905.4480 ] [http://inspirehep.net/search?p=find+J+JHEP,0909,123

Web End =INSPIRE ].

[46] J. Suresh and V.C. Kuriakose, Area spectrum and thermodynamics of KS black holes in Hoava gravity, http://dx.doi.org/10.1007/s10714-013-1565-2

Web End =Gen. Rel. Grav. 45 (2013) 1877 [http://arxiv.org/abs/1307.6438

Web End =arXiv:1307.6438 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.6438

Web End =INSPIRE ].

[47] J. Suresh and V.C. Kuriakose, Thermodynamics and quasinormal modes of Park black hole in Hoava gravity, http://dx.doi.org/10.1140/epjc/s10052-013-2613-5

Web End =Eur. Phys. J. C 73 (2013) 2613 [http://arxiv.org/abs/1310.2011

Web End =arXiv:1310.2011 ] [http://inspirehep.net/search?p=find+J+Eur.Phys.J.,C73,2613

Web End =INSPIRE ].

[48] H. Quevedo, Geometrothermodynamics, http://dx.doi.org/10.1063/1.2409524

Web End =J. Math. Phys. 48 (2007) 013506 [http://arxiv.org/abs/physics/0604164

Web End =physics/0604164 ] [http://inspirehep.net/search?p=find+J+J.Math.Phys.,48,013506

Web End =INSPIRE ].

[49] H. Quevedo, Geometrothermodynamics of black holes, http://dx.doi.org/10.1007/s10714-007-0586-0

Web End =Gen. Rel. Grav. 40 (2008) 971 [http://arxiv.org/abs/0704.3102

Web End =arXiv:0704.3102 ] [http://inspirehep.net/search?p=find+J+Gen.Rel.Grav.,40,971

Web End =INSPIRE ].

[50] H. Quevedo and A. Vazquez, The geometry of thermodynamics, in Recent developments in gravitation and cosmology, A. Macias et al. eds., AIP, New York U.S.A. (2008).

[51] P. Salamon, E. Ihrig and R.S. Berry, A group of coordinate transformations which preserve the metric of Weinhold, http_://dx.doi.org/10.1063/1.525629

Web End =J. Math. Phys. 24 (1983) 2515 .

[52] R. Mrugala, J.D. Nulton, J.C. Schon and P. Salamon, Statistical approach to the geometric structure of thermodynamics, http://dx.doi.org/10.1103/PhysRevA.41.3156

Web End =Phys. Rev. A 41 (1990) 3156 .

[53] R. Banerjee, S.K. Modak and S. Samanta, Glassy phase transition and stability in black holes, http://dx.doi.org/10.1140/epjc/s10052-010-1443-y

Web End =Eur. Phys. J. C 70 (2010) 317 [http://arxiv.org/abs/1002.0466

Web End =arXiv:1002.0466 ] [http://inspirehep.net/search?p=find+J+Eur.Phys.J.,C70,317

Web End =INSPIRE ].

[54] R. Banerjee, S.K. Modak and S. Samanta, Second order phase transition and thermodynamic geometry in Kerr-AdS black hole, http://dx.doi.org/10.1103/PhysRevD.84.064024

Web End =Phys. Rev. D 84 (2011) 064024 [http://arxiv.org/abs/1005.4832

Web End =arXiv:1005.4832 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D84,064024

Web End =INSPIRE ].

15

JHEP01(2015)019

[55] R. Banerjee and D. Roychowdhury, Critical phenomena in Born-Infeld AdS black holes, http://dx.doi.org/10.1103/PhysRevD.85.044040

Web End =Phys. Rev. D 85 (2012) 044040 [http://arxiv.org/abs/1111.0147

Web End =arXiv:1111.0147 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D85,044040

Web End =INSPIRE ].

[56] R. Banerjee, S. Ghosh and D. Roychowdhury, New type of phase transition in Reissner Nordstrom AdS black hole and its thermodynamic geometry, http://dx.doi.org/10.1016/j.physletb.2010.12.010

Web End =Phys. Lett. B 696 (2011) 156 [http://arxiv.org/abs/1008.2644

Web End =arXiv:1008.2644 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B696,156

Web End =INSPIRE ].

[57] R. Banerjee and D. Roychowdhury, Thermodynamics of phase transition in higher dimensional AdS black holes, http://dx.doi.org/10.1007/JHEP11(2011)004

Web End =JHEP 11 (2011) 004 [http://arxiv.org/abs/1109.2433

Web End =arXiv:1109.2433 ] [http://inspirehep.net/search?p=find+J+JHEP,1111,004

Web End =INSPIRE ].

[58] R. Banerjee, S.K. Modak and D. Roychowdhury, A unied picture of phase transition: from liquid-vapour systems to AdS black holes, http://dx.doi.org/10.1007/JHEP10(2012)125

Web End =JHEP 10 (2012) 125 [http://arxiv.org/abs/1106.3877

Web End =arXiv:1106.3877 ] [http://inspirehep.net/search?p=find+J+JHEP,1210,125

Web End =INSPIRE ].

[59] Th. M. Nieuwenhuizen, Ehrenfest relations at the glass transition: solution to an old paradox, http://dx.doi.org/10.1103/PhysRevLett.79.1317

Web End =Phys. Rev. Lett. 79 (1997) 1317 .

[60] Th.M. Nieuwenhuizen, Thermodynamic picture of the glassy state, http://dx.doi.org/10.1088/0953-8984/12/29/326

Web End =J. Phys. Condens. Matter http://dx.doi.org/10.1088/0953-8984/12/29/326

Web End =12 (2000) 6543 .

[61] J. Jackle, Models of the glass transition, http://dx.doi.org/10.1088/0034-4885/49/2/002

Web End =Rep. Prog. Phys. 49 (1986) 171 .[62] A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, http://dx.doi.org/10.1103/PhysRevD.60.064018

Web End =Phys. Rev. D 60 (1999) 064018 [http://arxiv.org/abs/hep-th/9902170

Web End =hep-th/9902170 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D60,064018

Web End =INSPIRE ].

[63] A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Holography, thermodynamics and uctuations of charged AdS black holes, http://dx.doi.org/10.1103/PhysRevD.60.104026

Web End =Phys. Rev. D 60 (1999) 104026 [http://arxiv.org/abs/hep-th/9904197

Web End =hep-th/9904197 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D60,104026

Web End =INSPIRE ].

[64] M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal eld theories, http://dx.doi.org/10.1088/0264-9381/17/2/310

Web End =Class. Quant. Grav. 17 (2000) 399 [http://arxiv.org/abs/hep-th/9908022

Web End =hep-th/9908022 ] [http://inspirehep.net/search?p=find+J+Class.Quant.Grav.,17,399

Web End =INSPIRE ].

[65] S. Nojiri and S.D. Odintsov, Anti-de Sitter black hole thermodynamics in higher derivative gravity and new conning deconning phases in dual CFT, http://dx.doi.org/10.1016/S0370-2693(01)01186-8

Web End =Phys. Lett. B 521 (2001) 87 [Erratum ibid. B 542 (2002) 301] [http://arxiv.org/abs/hep-th/0109122

Web End =hep-th/0109122 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B521,87

Web End =INSPIRE ].

[66] R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, http://dx.doi.org/10.1103/PhysRevD.65.084014

Web End =Phys. Rev. D 65 (2002) 084014 [http://arxiv.org/abs/hep-th/0109133

Web End =hep-th/0109133 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D65,084014

Web End =INSPIRE ].

[67] M. Cveti, S. Nojiri and S.D. Odintsov, Black hole thermodynamics and negative entropy in de Sitter and Anti-de Sitter Einstein-Gauss-Bonnet gravity, http://dx.doi.org/10.1016/S0550-3213(02)00075-5

Web End =Nucl. Phys. B 628 (2002) 295 [http://arxiv.org/abs/hep-th/0112045

Web End =hep-th/0112045 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0112045

Web End =INSPIRE ].

[68] S. Carlip and S. Vaidya, Phase transitions and critical behavior for charged black holes, http://dx.doi.org/10.1088/0264-9381/20/16/319

Web End =Class. Quant. Grav. 20 (2003) 3827 [http://arxiv.org/abs/gr-qc/0306054

Web End =gr-qc/0306054 ] [http://inspirehep.net/search?p=find+J+Class.Quant.Grav.,20,3827

Web End =INSPIRE ].

[69] R.-G. Cai and A. Wang, Thermodynamics and stability of hyperbolic charged black holes, http://dx.doi.org/10.1103/PhysRevD.70.064013

Web End =Phys. Rev. D 70 (2004) 064013 [http://arxiv.org/abs/hep-th/0406057

Web End =hep-th/0406057 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D70,064013

Web End =INSPIRE ].

[70] Y.S. Myung, No Hawking-Page phase transition in three dimensions, http://dx.doi.org/10.1016/j.physletb.2005.08.047

Web End =Phys. Lett. B 624 http://dx.doi.org/10.1016/j.physletb.2005.08.047

Web End =(2005) 297 [http://arxiv.org/abs/hep-th/0506096

Web End =hep-th/0506096 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B624,297

Web End =INSPIRE ].

[71] B.M.N. Carter and I.P. Neupane, Thermodynamics and stability of higher dimensional rotating (Kerr) AdS black holes, http://dx.doi.org/10.1103/PhysRevD.72.043534

Web End =Phys. Rev. D 72 (2005) 043534 [http://arxiv.org/abs/gr-qc/0506103

Web End =gr-qc/0506103 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D72,043534

Web End =INSPIRE ].

[72] R.-G. Cai, S.P. Kim and B. Wang, Ricci at black holes and Hawking-Page phase transition in Gauss-Bonnet gravity and dilaton gravity, http://dx.doi.org/10.1103/PhysRevD.76.024011

Web End =Phys. Rev. D 76 (2007) 024011 [http://arxiv.org/abs/0705.2469

Web End =arXiv:0705.2469 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D76,024011

Web End =INSPIRE ].

[73] Y.S. Myung, Y.-W. Kim and Y.-J. Park, Thermodynamics and phase transitions in the Born-Infeld-Anti-de Sitter black holes, http://dx.doi.org/10.1103/PhysRevD.78.084002

Web End =Phys. Rev. D 78 (2008) 084002 [http://arxiv.org/abs/0805.0187

Web End =arXiv:0805.0187 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D78,084002

Web End =INSPIRE ].

16

JHEP01(2015)019

[74] Y.S. Myung, Phase transition between non-extremal and extremal Reissner-Nordstrom black holes, http://dx.doi.org/10.1142/S0217732308026741

Web End =Mod. Phys. Lett. A 23 (2008) 667 [http://arxiv.org/abs/0710.2568

Web End =arXiv:0710.2568 ] [http://inspirehep.net/search?p=find+J+Mod.Phys.Lett.,A23,667

Web End =INSPIRE ].

[75] G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Phase transitions in charged topological-AdS black holes, http://dx.doi.org/10.1088/1126-6708/2008/05/107

Web End =JHEP 05 (2008) 107 [http://arxiv.org/abs/0801.4921

Web End =arXiv:0801.4921 ] [http://inspirehep.net/search?p=find+J+JHEP,0805,107

Web End =INSPIRE ].

[76] M. Cadoni, G. DAppollonio and P. Pani, Phase transitions between Reissner-Nordstrom and dilatonic black holes in 4D AdS spacetime, http://dx.doi.org/10.1007/JHEP03(2010)100

Web End =JHEP 03 (2010) 100 [http://arxiv.org/abs/0912.3520

Web End =arXiv:0912.3520 ] [http://inspirehep.net/search?p=find+J+JHEP,1003,100

Web End =INSPIRE ].

[77] H. Liu, H. L, M. Luo and K.-N. Shao, Thermodynamical metrics and black hole phase transitions, http://dx.doi.org/10.1007/JHEP12(2010)054

Web End =JHEP 12 (2010) 054 [http://arxiv.org/abs/1008.4482

Web End =arXiv:1008.4482 ] [http://inspirehep.net/search?p=find+J+JHEP,1012,054

Web End =INSPIRE ].

[78] A. Sahay, T. Sarkar and G. Sengupta, On the phase structure and thermodynamic geometry of R-charged black holes, http://dx.doi.org/10.1007/JHEP11(2010)125

Web End =JHEP 11 (2010) 125 [http://arxiv.org/abs/1009.2236

Web End =arXiv:1009.2236 ] [http://inspirehep.net/search?p=find+J+JHEP,1011,125

Web End =INSPIRE ].

[79] Q.-J. Cao, Y.-X. Chen and K.-N. Shao, Black hole phase transitions in Hoava-Lifshitz gravity, http://dx.doi.org/10.1103/PhysRevD.83.064015

Web End =Phys. Rev. D 83 (2011) 064015 [http://arxiv.org/abs/1010.5044

Web End =arXiv:1010.5044 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D83,064015

Web End =INSPIRE ].

[80] S.-W. Wei, Y.-X. Liu, Y.-Q. Wang and H. Guo, Thermodynamic geometry of black hole in the deformed Hoava-Lifshitz gravity, http://dx.doi.org/10.1209/0295-5075/99/20004

Web End =Europhys. Lett. 99 (2012) 20004 [http://arxiv.org/abs/1002.1550

Web End =arXiv:1002.1550 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.1550

Web End =INSPIRE ].

[81] B.R. Majhi and D. Roychowdhury, Phase transition and scaling behavior of topological charged black holes in Hoava-Lifshitz gravity, http://dx.doi.org/10.1088/0264-9381/29/24/245012

Web End =Class. Quant. Grav. 29 (2012) 245012 [http://arxiv.org/abs/1205.0146

Web End =arXiv:1205.0146 ] [http://inspirehep.net/search?p=find+J+Class.Quant.Grav.,29,245012

Web End =INSPIRE ].

[82] W. Kim and Y. Kim, Phase transition of quantum corrected Schwarzschild black hole, http://dx.doi.org/10.1016/j.physletb.2012.11.017

Web End =Phys. http://dx.doi.org/10.1016/j.physletb.2012.11.017

Web End =Lett. B 718 (2012) 687 [http://arxiv.org/abs/1207.5318

Web End =arXiv:1207.5318 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B718,687

Web End =INSPIRE ].

[83] Y.-D. Tsai, X.N. Wu and Y. Yang, Phase structure of Kerr-AdS black hole, http://dx.doi.org/10.1103/PhysRevD.85.044005

Web End =Phys. Rev. D 85 http://dx.doi.org/10.1103/PhysRevD.85.044005

Web End =(2012) 044005 [http://arxiv.org/abs/1104.0502

Web End =arXiv:1104.0502 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D85,044005

Web End =INSPIRE ].

[84] F. Capela and G. Nardini, Hairy black holes in massive gravity: thermodynamics and phase structure, http://dx.doi.org/10.1103/PhysRevD.86.024030

Web End =Phys. Rev. D 86 (2012) 024030 [http://arxiv.org/abs/1203.4222

Web End =arXiv:1203.4222 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D86,024030

Web End =INSPIRE ].

[85] D. Kubiznak and R.B. Mann, P-V criticality of charged AdS black holes, http://dx.doi.org/10.1007/JHEP07(2012)033

Web End =JHEP 07 (2012) http://dx.doi.org/10.1007/JHEP07(2012)033

Web End =033 [http://arxiv.org/abs/1205.0559

Web End =arXiv:1205.0559 ] [http://inspirehep.net/search?p=find+J+JHEP,1207,033

Web End =INSPIRE ].

[86] S.-W. Wei and Y.-X. Liu, Critical phenomena and thermodynamic geometry of charged Gauss-Bonnet AdS black holes, http://dx.doi.org/10.1103/PhysRevD.87.044014

Web End =Phys. Rev. D 87 (2013) 044014 [http://arxiv.org/abs/1209.1707

Web End =arXiv:1209.1707 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D87,044014

Web End =INSPIRE ].

[87] M. Eune, W. Kim and S.-H. Yi, Hawking-Page phase transition in BTZ black hole revisited, http://dx.doi.org/10.1007/JHEP03(2013)020

Web End =JHEP 03 (2013) 020 [http://arxiv.org/abs/1301.0395

Web End =arXiv:1301.0395 ] [http://inspirehep.net/search?p=find+J+JHEP,1303,020

Web End =INSPIRE ].

[88] P.C.W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978) 1313.

17

JHEP01(2015)019