Temperature of McVittie Black Holes in Cavities

Abstract: In this paper, we study the temperature of the McVittie black holes in cavities. The cavities are isothermal boxes that can make the McVittie black holes thermodynamically stable. The McVittie black holes are dynamical black holes, so the traditional action method can hardly be used to calculate the cavity temperature. We find that the cavity temperature $T_c$ can be derived from the McVittie black hole temperature $T_+$ and the Tolman relation. By setting $H^2$ to be $\Lambda/3$, we also get the cavity temperature of de Sitter black holes, which are interestingly the same with the results from the action method, but our method is much more easier.