To understand the effect of third order Lovelock gravity, P-V criticality of topological AdS black holes in Lovelock-Born-Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and in some more detail than the previous literature. A detailed analysis of the limit case β → ∞ is performed for the sevendimensional black holes. It is shown that, for the spherical topology, P-V criticality exists for both the uncharged and the charged cases. Our results demonstrate again that the charge is not the indispensable condition of P-V criticality. It may be attributed to the effect of higher derivative terms of the curvature because similar phenomenon was also found for Gauss-Bonnet black holes. For k = 0, there would be no P-V criticality. Interesting findings occur in the case k = −1, in which positive solutions of critical points are found for both the uncharged and the charged cases. However, the P-v diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of α, the entropy is always positive for any specific volume v. Since no P-V criticality exists for k = −1 in Einstein gravity and Gauss-Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss-Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and "swallow tail" behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research. a

Effects of the dimensionality on the Joule-Thomson expansion are discussed in detail by considering the case of d-dimensional charged AdS black holes. Specifically, we investigate three important aspects characteristic of the Joule-Thomson expansion. Namely, the Joule-Thomson coefficient, the inversion curves and the isenthalpic curves. We utilize two different approaches to derive the explicit expression of the Joule-Thomson coefficient and show that both approaches are consistent with each other. The divergent point and the zero point of the Joule-Thomson coefficient are discussed. The former is shown to reveal the information of Hawking temperature while the latter is depicted through the so-called inversion curves. Fine structures of the inversion curves are disclosed in the cases d > 4. At low pressure, the inversion temperature increases with the dimensionality d while at high pressure it decreases with d. The ratio between minimum inversion temperature Tmin and the critical temperature Tc is discussed with its explicit expression obtained for d > 4. Surprisingly, it is shown that the ratio is not always equal to 1/2 but decreases with the dimensionality d. Moreover, isenthalpic curves of d > 4 are shown to expand toward higher pressure when the dimensionality d increases.

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Following an earlier study regarding EinsteinGauss-Bonnet-massive black holes in the presence of a Born-Infeld nonlinear electromagnetic field (Hendi, arXiv:1510.00108, 2016, we study thermodynamical structure and critical behavior of these black holes through various methods in this paper. Geometrical thermodynamics is employed to give a picture regarding the phase transition of these black holes. Next, a new method is used to derive critical pressure and radius of the horizon of these black holes. In addition, Maxwell equal area law is employed to study the Van der Waals like behavior of these black holes. Moreover, the critical exponents are calculated and by using Ehrenfest equations, the type of phase transition is determined.

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Aiming at a unified phase transition picture of the charged topological black hole in Hořava-Lifshitz gravity, we investigate this issue not only in canonical ensemble with the fixed charge case but also in grand-canonical ensemble with the fixed potential case. We firstly perform the standard analysis of the specific heat, the free energy and the Gibbs potential, and then study its geometrothermodynamics. It is shown that the local phase transition points not only witness the divergence of the specific heat, but also witness the minimum temperature and the maximum free energy or Gibbs potential. They also witness the divergence of the corresponding thermodynamic scalar curvature. No matter which ensemble is chosen, the metric constructed can successfully produce the behavior of the thermodynamic interaction and phase transition structure while other metrics failed to predict the phase transition point of the charged topological black hole in former literature. In grand-canonical ensemble, we have discovered the phase transition which has not been reported before. It is similar to the canonical ensemble in which the phase transition only takes place when k = −1. But it also has its unique characteristics that the location of the phase transition point depends on the value of potential, which is different from the canonical ensemble where the phase transition point is independent of the parameters. After an analytical check of Ehrenfest scheme, we find that the new phase transition is a second order one. It is also found that the thermodynamics of the black hole in Horava-Lifshitz gravity is quite different from that in Einstein gravity. 1

To provide an analytic verification of the nature of phase transition at the critical point of P − V criticality, the original expressions of Ehrenfest equations have been introduced directly. By treating the cosmological constant and its conjugate quantity as thermodynamic pressure and volume respectively, we carry out analytical check of classical Ehrenfest equations. To show that our approach is universal, we investigate not only higher-dimensional charged AdS black holes, but also rotating AdS black holes. Not only are the examples of Einstein gravity shown, but also the example of modified gravity is presented for Gauss-Bonnet AdS black holes. The specific heat at constant pressure CP , the volume expansion coefficient α and the isothermal compressibility coefficient κT are found to diverge exactly at the critical point. It has been verified that both Ehrenfest equations hold at the critical point of P − V criticality in the extended phase spaces of AdS black holes. So the nature of the critical point of P − V criticality of AdS black holes has been demonstrated analytically to be a second-order phase transition. These results are consistent with the nature of liquid-gas phase transition at the critical point. In this sense, our research would deepen the understanding of the relations of AdS black holes and liquid-gas systems. Moreover, our successful approaches to introduce the original expressions of Erhenfest equations directly into black hole phase transition research demonstrate again that black hole thermodynamics is closely related to classical thermodynamics, which allows us to borrow techniques from classical thermodynamics to investigate the thermodynamics of black holes.

A consistent and unified picture for critical phenomena of charged AdS black holes in f (R) gravity is drawn in this paper. Firstly, we investigate the phase transition in canonical ensemble. We derive the explicit solutions corresponding to the divergence of C Q . The two solutions merge into one when the condition Q c = −1 3R 0 is satisfied. The curve of specific heat for Q < Q c has two divergent points and can be divided into three regions. Both the large radius region and the small radius region are thermodynamically stable with positive specific heat while the medium radius region is unstable with negative specific heat. However, when Q > Q c , the specific heat is always positive, implying the black holes are locally stable and no phase transition will take place. Secondly, both the T − r + curve and T − S curve f (R) AdS black holes are investigated and they exhibit Van der Vaals like behavior as the P − v curve in the former research. Critical physical quantities are obtained and they are consistent with those derived from the specific heat analysis. We carry out numerical check of Maxwell equal area law for the cases Q = 0.2Q c , 0.4Q c , 0.6Q c , 0.8Q c . The relative errors are amazingly small and can be negligible. So the Maxwell equal area law holds for T − S curve of f (R) black holes. Thirdly, we establish geometrothermodynamics for f (R) AdS black hole to examine the phase structure. It is shown that the Legendre invariant scalar curvature R would diverge exactly where the specific heat diverges. To summarize, the above three perspectives are consistent with each other, thus providing a unified picture which deepens the understanding of critical phenomena of f (R) AdS black holes.