De Sitter entropy: on-shell versus off-shell

Attributing thermodynamic properties to the Bunch-Davies state in static patch of de Sitter space and setting the corresponding equations of state, we demonstrate that, for pure gravity, the bulk entropy computed on-shell as a volume integral in de Sitter space coincides with the Wald entropy (area law) in any spacetime dimension and for any theory of f® gravity. We extend this result to the renormalized entanglement entropy of a non-minimally coupled scalar field. From the on-shell perspective, entropy emerges as a bulk contribution, whereas from the off-shell viewpoint, it manifests as a boundary (horizon) contribution. As a result, in de Sitter space, generalized entropy can be understood in two distinct ways: either as a bulk or as a boundary contribution.