In this thesis, we investigate the thermodynamics of the de Sitter static patch in order to extract information which can constrain microscopic models of de Sitter quantum gravity. We begin by reviewing previous works which demonstrate how to make sense of the seemingly ill-defined static patch density of states through the introduction of Harish-Chandra group characters, or equivalently through renormalization with respect to a reference problem in Rindler space. A thermal partition function can then be constructed and expressed in terms of a sum over quasinormal mode frequencies. We recap how, in the scalar case, this partition function is equivalent to a 1-loop sphere path integral, as expected from the Gibbons-Hawking proposal, and provides macroscopic data which microscopic models must be consistent with. We next present novel results dealing with scalar Green functions in de Sitter. After constructing various static patch correlators and showing how they can be obtained from their sphere counterparts, we relate the spectral Green function to the Harish-Chandra characters that we came across before, tying them to observables directly accessible within the static patch. We comment on how this result will allow us to generalize thermodynamic considerations to interacting theoriesand therefore place stronger consistency constraints on microscopic models. We finally generalize our analysis to spinning fields, for which thermal partition functions differ from Euclidean path integrals by edge corrections. We reveal new findings which trace the source of these discrepancies to those quasinormal modes which do not correspond to regular Euclidean solutions, explicitly demonstrating this through several examples. Our results highlight the differences between Lorentzian and Euclidean approaches to de Sitter thermodynamics, and hint at new avenues to pursue in the hopes of providing more consistency constraints.

In order to find quantum corrections to the de Sitter entropy, a new approach to higher loop Feynman integral computations on the sphere is presented. Arbitrary scalar Feynman integrals on a spherical background are brought into the generalized Euler integral (A-hypergeometric series/GKZ systems) form by expressing the massive scalar propagator as a bivariate radial Mellin transform of the massless scalar propagator in one higher dimensional Euclidean flat space. This formulation is expanded to include massive and massless vector fields by construction of similar embedding space propagators. Vector Feynman integrals are shown to be sums over generalized Euler integral formed of underlying scalar Feynman integrals. Granting existence of general spin embedding space propagators, general spin Feynman integrals are shown, by the construction of a "master" integral, to also be sums over generalized Euler integral representations of scalar Feynman integrals. Finding exact embedding space propagator expressions for fields of integer spin ≥ 2 and half integer spin is left to future work.