De Sitter Entropy and the Gravitational Path Integral

<!--HTML--><p>Gibbons and Hawking famously derived the de Sitter horizon entropy from the Euclidean gravitational path integral in the saddle point approximation. In this talk we extend this result in two different ways. First, we compute the entropy of de Sitter black holes from the on-shell Euclidean action, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. We use this to calculate the pair creation rate of arbitrary mass black holes in de Sitter space. Second, in two-dimensional de Sitter space we show how the generalized entropy can be obtained from an on-shell action for semiclassical dilaton gravity in the microcanonical ensemble. Minimizing the action yields extremizing the generalized entropy, consistent with the island formula.</p>