Cosmological quantum states of de Sitter-Schwarzschild are static patch partition functions

We solve the Wheeler-DeWitt equation in the 'cosmological interior' (the past causal diamond of future infinity) of four dimensional dS-Schwarzschild spacetimes. Within minisuperspace there is a basis of solutions labelled by a constant $c$, conjugate to the mass of the black hole. We propose that these solutions are in correspondence with partition functions of a dual quantum mechanical theory where $c$ plays the role of time. The quantum mechanical theory lives on worldtubes in the 'static patch' of dS-Schwarzschild, and the partition function is obtained by evolving the corresponding Wheeler-DeWitt wavefunction through the cosmological horizon, where a metric component $g_{tt}$ changes sign. We establish that the dual theory admits a symmetry algebra given by a central extension of the Poincar\'e algebra $\mathfrak{e}(1,1)$ and that the entropy of the dS black hole is encoded as an averaging of the dual partition function over the background $g_{tt}$.