Thermodynamics of black holes in massive gravity

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) >= 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.