Black-hole thermodynamics and the Euclidean Einstein action

Using the approach to black-hole thermodynamics initiated by Gibbons and Hawking, in terms of the Euclidean Einstein action, I show that the canonical ensemble with elements of radius r and temperature T(r) for hot gravity with black holes is well defined. This follows from the double valuedness of solutions of the Euclidean Einstein equation with canonical boundary conditions. One of the solutions is a locally stable hole. Its partition function is well defined and implies the entropy S=4π$M^2$ as well as a generalized version of black-hole thermodynamics that reduces to the usual form if ${\mathrm{rM}}^{\mathrm{-}1}$→∞. The density of states of the locally stable hole is real and nonpathological. The free energy of this hole can be negative, while that of the other (unstable) solution is always positive. Consequently, the direct nucleation of black holes from hot flat space, as proposed by Gross, Perry, and Yaffe, can be given a thermodynamically consistent description. The scaling laws for hot gravity are obtained and applied to phase transitions between hot flat space and locally stable holes. The free energy of the unstable solution forms the effective potential barrier between these phases. The ground state of the canonical ensemble is always locally stable in the semiclassical approximation. If N is the effective number of massless fields of helicity zero in hot flat space, then when either r≲$N^{1/2}$ or T≳$N^{\mathrm{-}1/2}$, hot flat space is the most probable ground state. Independently of N, if rT${<\left(27\right)\mathrm{}}^{1/2}$(8π${)}^{\mathrm{-}1}$ there can be no real black hole in the canonical ensemble.