Title: de Sitter state as thermal bath

Abstract: We consider the local thermodynamics of the de Sitter state. The local temperature, $T=H/\pi$, where $H$ is the Hubble parameter, is twice larger than the Gibbons-Hawking temperature of cosmological horizon, $T_{\rm GH}=H/2\pi$. The local temperature determines the rate of the activation processes in the de Sitter environment, such as the process of the ionization of atom. The local temperature determines the local entropy of the de Sitter vacuum state, and this allows to calculate the total entropy inside the cosmological horizon. The result reproduces the Gibbons-Hawking area law, which is related to the cosmological horizon, $S_{\rm hor}=4\pi KA$, where $K=1/16\pi G$. We extend the consideration of the local thermodynamics of the de Sitter state using the $f({\cal R})$ gravity. In this thermodynamics, the Ricci scalar curvature ${\cal R}$ and the effective gravitational coupling, $K$, are thermodynamically conjugate variables. The holographic connection between the bulk entropy of the Hubble volume, and the surface entropy of the cosmological horizon remains the same, $S_{\rm hor}=4\pi KA$, but where the gravitational coupling is $K=df/d{\cal R}$. The local temperature of the de Sitter vacuum suggests that it serves as the heat bath for the matter, and thus the de Sitter state is locally unstable towards the creation of matter and its further heating. The decay of the de Sitter vacuum due to such processes determines the quantum breaking time of the space-times with positive cosmological constant. We also consider the thermodynamics of de Sitter in the frame of the multi-metric gravity ensemble, where the heat exchange between different "sub-Universes" in the ensemble leads to the common de Sitter expansion with common temperature $T=H/\pi$. Application of the local thermodynamics to the entropy of the Schwarzschild black hole is also considered.