In this Letter, we investigate the effects of dark energy on P − V criticality of charged AdS black holes by considering the case of the RN-AdS black holes surrounded by quintessence. By treating the cosmological constant as thermodynamic pressure, we study its thermodynamics in the extended phase space. It is shown that quintessence dark energy does not affect the existence of small/large black hole phase transition. For the case ω q = −2/3 we derive analytic expressions of critical physical quantities, while for cases ω q = −2/3 we appeal to numerical method for help. It is shown that quintessence dark energy affects the critical physical quantities near the critical point. Critical exponents are also calculated. They are exactly the same as those obtained before for arbitrary other AdS black holes, which implies that quintessence dark energy does not change the critical exponents.

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.

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

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.

In this paper, we investigate the coexistence curves and molecule number densities of f (R) AdS black holes and Gauss-Bonnet AdS black holes. Specifically, we work with the reduced parameter space and derive the analytic expressions of the universal coexistence curves that are independent of theory parameters. Moreover, we obtain the explicit expressions of the physical quantity describing the difference of the number densities of black hole molecules between the small and large black hole. It is found that both the coexistence curve and the difference of the molecule number densities of f (R) AdS black holes coincide with those of RN-AdS black holes. It may be attributed to the same equation of state they share in the reduced parameter space. The difference of the molecule number densities between the small and large Gauss-Bonnet AdS black hole exhibits different behavior. This may be attributed to the fact that the charge of RN-AdS black hole is non-trivial. Our research will not only deepen the understanding of both the physics along the coexistence curve and the underlying microscopic freedom of AdS black holes, but also highlight the importance of the law of corresponding states.

Effects of Lovelock gravity on the Joule-Thomson expansion are probed from various perspectives. The well-known Joule-Thomson coefficient is derived with both the explicit expression and intuitive image presented. Moreover, the inversion curves showing the relation between the inversion temperature and the inversion pressure are studied. It is shown that for given inversion pressure, the inversion temperature of the case α = 0 (α is the Lovelock parameter) is much lower than that of the case α = 0. And the inversion temperature tends to decrease with α, in contrast to the effect of the electric charge. It is also shown that the ratio between the minimum inversion temperature and the critical temperature decreases with α for α = 0. Furthermore, the isenthalpic curves are investigated with rich physics revealed. The intersection point between the isenthalpic curve and the inversion curve is exactly the inversion point discriminating the heating process from cooling process. It is shown that both the inversion temperature and the inversion pressure for α = 0 are much lower for the same given mass of the black hole, showing the effect of Lovelock gravity. Last but not the least, we discuss the case of uncharged Lovelock AdS black holes with interesting feature found. It is shown that the Joule-Thomson coefficient is always positive, suggesting the expansion is always in the regime of cooling process. And no inversion temperature exists, in contrast to the case Q = 0. Isenthalpic curves are also quite different since the temperature increases monotonically with the pressure when the mass is specified.