Schwarzschild-de Sitter black holes have two horizons that are at different temperatures for generic values of the black hole mass. Since the horizons are out of equilibrium the solutions do not admit a smooth Euclidean continuation. Using the factorization properties of the path integral I will show that Euclidean SdS is a genuine saddle point in the presence of a constraint, providing a consistent Euclidean computation of the probability to find a black hole in the de Sitter bath.
