JETP Letters: issues online

We consider the local thermodynamics of the de Sitter state in the $f({\cal R})$ gravity. The local temperature, which is the same for all points of the de Sitter space, is T=H/π, where H is the Hubble parameter. It is twice larger than the Gibbons-Hawking temperature of the cosmological horizon, T_GH=H/2π. The local temperature is not related to the cosmological horizon. It determines the rate of the activation processes, which are possible in the de Sitter environment. The typical example is the process of the ionization of the atom in the de Sitter environment, which rate is determined by temperature T=H/π. 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 corresponds to the Wald entropy, S_hor=4π KA. Here K is the effective gravitational coupling, $K=df/d{\cal R}$ . In the local thermodynamic approach, K is the thermodynamic variable, which is conjugate to the Ricci scalar curvature ${\cal R}$ . The holographic connection between the bulk entropy of the Hubble volume and the surface entropy of the cosmological horizon supports the suggestion that the de Sitter quantum vacuum is characterized by the local thermodynamics with the local temperature T=H/π. The local temperature T=H/π of the de Sitter vacuum suggests that the de Sitter vacuum 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.