In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law firstly proposed by Hayward to treat the "outer"trapping horizon of dynamical black hole can be used to the apparent horizon (a kind of "inner" trapping horizon in the context of the FRW cosmology) of the FRW universe. We discuss three kinds of gravity theorties: Einstein theory, Lovelock thoery and scalar-tensor theory. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon. In Lovelock theory, treating the higher derivative terms as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. In scalar-tensor gravity, we find, by using the same procedure, the Clausius relation no longer holds. This indicates that the apparent horizon of FRW universe in the scalar-tensor gravity corresponds to a system of non-equilibrium thermodynamics. We show this point by using the method developed recently by Eling et al. for dealing with the f (R) gravity.

We study the P − V criticality and phase transition in the extended phase space of charged Gauss-Bonnet black holes in anti-de Sitter space, where the cosmological constant appears as a dynamical pressure of the system and its conjugate quantity is the thermodynamic volume of the black holes. The black holes can have a Ricci flat (k = 0), spherical (k = 1), or hyperbolic (k = −1) horizon. We find that for the Ricci flat and hyperbolic Gauss-Bonnet black holes, no P − V criticality and phase transition appear, while for the black holes with a spherical horizon, even when the charge of the black hole is absent, the P − V criticality and the small black hole/large black hole phase transition will appear, but it happens only in d = 5 dimensions; when the charge does not vanish, the P − V criticality and the small black hole/large phase transition always appear in d = 5 dimensions; in the case of d ≥ 6, to have the P − V criticality and the small black hole/large black hole phase transition, there exists an upper bound for the parameter b = α|Q| −2/(d−3) , whereα is the Gauss-Bonnet coefficient and Q is the charge of the black hole. We calculate the critical exponents at the critical point and find that for all cases, they are the same as those in the van der Waals liquid-gas system. *

Hawking radiation is an important quantum phenomenon of black hole, which is closely related to the existence of event horizon of black hole. The cosmological event horizon of de Sitter space is also of the Hawking radiation with thermal spectrum. By use of the tunneling approach, we show that there is indeed a Hawking radiation with temperature, T = 1/2πr A , for locally defined apparent horizon of a Friedmann-Robertson-Walker universe with any spatial curvature, wherer A is the apparent horizon radius. Thus we fill in the gap existing in the literature investigating the relation between the first law of thermodynamics and Friedmann equations, there the apparent horizon is assumed to have such a temperature without any proof. In addition, we stress the implication of the Hawking temperature associated with the apparent horizon.

In this note by use of the holographic principle together with the equipartition law of energy and the Unruh temperature, we derive the Friedmann equations of a FriedmannRobertson-Walker universe. *

We study the P -V criticality and phase transition in the extended phase space of charged anti-de Sitter black holes in canonical ensemble of ghost-free massive gravity, where the cosmological constant is viewed as a dynamical pressure of the black hole system. We give the generalized thermodynamic first law and the Smarr relation with massive gravity correction. We find that not only when the horizon topology is spherical but also in the Ricci flat or hyperbolic case, there appear the P -V criticality and phase transition up to the combination k+c 2 0 c2m 2 in the four-dimensional case, where k characterizes the horizon curvature and c2m 2 is the coefficient of the second term of massive potential associated with the graviton mass. The positivity of such combination indicate the van der Waals-like phase transition. When the spacetime dimension is larger then four, the Maxwell charge there seems unnecessary for the appearance of critical behavior, but a infinite repulsion effect needed, which can also be realized through negative valued c3m 2 or c4m 2 , which is third or fourth term of massive potential. When c3m 2 is positive, a Hawking-Page like black hole to vacuum phase transition is shown in the five-dimensional chargeless case. For the van der Waals like phase transition in four and five spacetime dimensions, we calculate the critical exponents near critical point and find they are the same as those in the van der Waals liquid-gas system.

We study the Hawking radiation of the spherically symmetric, asymptotically flat black holes in the infrared modified Hořava-Lifshitz gravity by applying the methods of covari-ant anomaly cancellation and effective action, as well as the approach of Damour-Ruffini-Sannan's. These black holes behave as the usual Schwarzschild ones of the general relativity when the radial distance is very large. We also extend the method of covariant anomaly cancellation to derive the Hawking temperature of the spherically symmetric, asymptotically AdS black holes that represent the analogues of the Schwarzschild AdS ones.

In this paper we discuss thermodynamics of apparent horizon of an n-dimensional Friedmann-Robertson-Walker (FRW) universe embedded in an (n+1)-dimensional AdS spacetime. By using the method of unified first law, we give the explicit entropy expression of the apparent horizon of the FRW universe. In the large horizon radius limit, this entropy reduces to the n-dimensional area formula, while in the small horizon radius limit, it obeys the (n + 1)-dimensional area formula. We also discuss the corresponding bulk geometry and study the apparent horizon extended into the bulk. We calculate the entropy of this apparent horizon by using the area formula of the (n + 1)-dimensional bulk. It turns out that both methods give the same result for the apparent horizon entropy. In addition, we show that the Friedmann equation on the brane can be rewritten to a form of the first law, dE = T dS + W dV , at the apparent horizon.

We study generalized Misner-Sharp energy in f (R) gravity in a spherically symmetric spacetime. We find that unlike the cases of Einstein gravity and Gauss-Bonnet gravity, the existence of the generalized Misner-Sharp energy depends on a constraint condition in the f (R) gravity. When the constraint condition is satisfied, one can define a generalized Misner-Sharp energy, but it cannot always be written in an explicit quasi-local form. However, such a form can be obtained in a FRW universe and for static spherically symmetric solutions with constant scalar curvature. In the FRW universe, the generalized Misner-Sharp energy is nothing but the total matter energy inside a sphere with radius r, which acts as the boundary of a finite region under consideration. The case of scalar-tensor gravity is also briefly discussed.