We present a class of charged black hole solutions in an ($n+2$)-dimensional massive gravity with a negative cosmological constant, and study the thermodynamics and phase structure of the black hole solutions in both the grand canonical and canonical ensembles. The black hole horizon can have a positive, zero, or negative constant curvature characterized by the constant $k$. By using the Hamiltonian approach, we obtain conserved charges of the solutions and find that the black hole entropy still obeys the area formula and the gravitational field equation at the black hole horizon can be cast into a form similar to the first law of black hole thermodynamics. In the grand canonical ensemble, we find that the thermodynamics and phase structure depend on the combination $k-{\mu }^2/4+c_2{m}^2$ in the four-dimensional case, where $\mu$ is the chemical potential and $c_2{m}^2$ is the coefficient of the second term in the potential associated with the graviton mass. When it is positive, the Hawking-Page phase transition can happen; when as it is negative, the black hole is always thermodynamically stable with a positive capacity. In the canonical ensemble, the combination turns out to be $k+c_2{m}^2$ in the four-dimensional case. When it is positive, a first-order phase transition can happen between small and large black holes if the charge is less than its critical value. In the higher-dimensional [$(n+2)\ge 5$] case, even when the charge is absent, the small/large black hole phase transition can also appear, and the coefficients for the third ($c_3{m}^2$) and/or fourth ($c_4{m}^2$) terms in the potential associated with the graviton mass in massive gravity can play the same role as that of the charge in the four-dimensional case.

We study the $P\text{−}V$ 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 $P\text{−}V$ criticality and phase transition up to the combination $k+c_0^2{c}_2{m}^2$ in the four-dimensional case, where $k$ characterizes the horizon curvature and $c_2{m}^2$ 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 $c_3{m}^2$ or $c_4{m}^2$, which is third or fourth term of massive potential. When $c_3{m}^2$ 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 the critical point and find they are the same as those in the van der Waals liquid-gas system.

We present a kind of generalized Vaidya solutions in a generic Lovelock gravity. This solution generalizes the simple case in Gauss-Bonnet gravity reported recently by some authors. We study the thermodynamics of apparent horizon in this generalized Vaidya spacetime. Treating those terms except for the Einstein tensor as an effective energy-momentum tensor in the gravitational field equations, and using the unified first law in Einstein gravity theory, we obtain an entropy expression for the apparent horizon. We also obtain an energy expression of this spacetime, which coincides with the generalized Misner-Sharp energy proposed by Maeda and Nozawa in Lovelock gravity.Comment: v3: comments and references added, some type errors corrected. Revtex4, 22 pages, no figure. to appear in Phys. Rev.

We study generalized Misner-Sharp energy in $f(R)$ gravity in a spherically symmetric space-time. 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 quasilocal form. However, such a form can be obtained in a Friedmann-Robertson-Walker universe and for static spherically symmetric solutions with constant scalar curvature. In the Friedmann-Robertson-Walker 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.