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We study the semi-classical thermodynamics of two-dimensional de Sitter space($\text{dS}_{2}$) in Jackiw-Teitelboim (JT) gravity coupled to conformalmatter. We extend the quasi-local formalism of Brown and York to$\text{dS}_{2}$, where a timelike boundary is introduced in the static patch touniquely define conserved charges, including quasi-local energy. The boundarydivides the static patch into two systems, a cosmological system and a blackhole system, the former being unstable under thermal fluctuations while thelatter is stable. A semi-classical quasi-local first law is derived, where theGibbons--Hawking entropy is replaced by the generalized entropy. In themicrocanonical ensemble the generalized entropy is stationary. Further, we showthe on-shell Euclidean microcanonical action of a causal diamond insemi-classical JT gravity equals minus the generalized entropy of the diamond,hence extremization of the entropy follows from minimizing the action. Thus, weprovide a first principles derivation of the island rule for $U(1)$ symmetric$\text{dS}_{2}$ backgrounds, without invoking the replica trick. We discuss theimplications of our findings for static patch de Sitter holography.

Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity