The generalized entropy in holographic semiclassical de Sitter spacetime

We consider generalized de Sitter entropy which includes both entanglement entropy of the strongly coupled quantum field and the gravitational Wald entropy in the framework of holographic semiclassical gravity. The quantum corrected de Sitter spacetime is constructed perturbatively by considering static deformations of the classical solution. By applying the Noether charge formula to our semiclassical model, we show that both the renormalized entropies remain finite without introducing a cutoff, even when the bulk AdS gravitational field does not satisfy the Dirichlet boundary condition. We show that the first order variation of the generalized entropy with respect to the de Sitter horizon always vanishes when the conformal boundary metric is promoted to a dynamical variable.