Path integrals for causal diamonds and the covariant entropy principle

We study causal diamonds in Minkowski, Schwarzschild, (anti-)de Sitter, and Schwarzschild-de Sitter spacetimes using Euclidean methods. The null boundaries of causal diamonds are shown to map to isolated punctures in the Euclidean continuation of the parent manifold. Boundary terms around these punctures decrease the Euclidean action by A(lozenge)/4, where A(lozenge) is the area of the holographic screen around the diamond. We identify these boundary contributions with the maximal entropy of gravitational degrees of freedom associated with the diamond.