  Schwarzschild-de Sitter (SdS) black holes do not admit a completely smooth
Euclidean continuation. We discuss some modifications of the gravitational path
integral that give Euclidean SdS a semiclassical equilibrium interpretation.
First we consider "gravity in a cavity," defining the canonical ensemble in a
box that excises one horizon. However, this standard approach does not work for
positive cosmological constant: the solution of lowest free energy has a
negative heat capacity, which is inconsistent if it is to provide the leading
semiclassical contribution to a canonical partition function. Instead we modify
the boundary conditions in the path integral to construct the microcanonical
partition function, which appears to be well-defined. We then bring two
ensembles into contact and remove the boundary, producing states of a larger
microcanonical ensemble that contain, for example, both a black hole and a
cosmological horizon at once. These systems are closed and have no boundary,
but they must possess some form of mild metric discontinuity. We discuss the
case where the discontinuity is equivalent to the insertion of a thin, rigid
membrane, separating two systems that can exchange energy and are at local
equilibrium. Equilibrium configurations obtained in this way are found to be
thermodynamically unstable if they contain a black hole.
