Analog Sommerfeld law in quantum vacuum

The activation temperature $T$ in the de Sitter environment is twice larger than the Gibbons-Hawking temperature, related to the cosmological horizon. We consider the activation temperature as the local temperature of the de Sitter vacuum, and construct the local thermodynamics of the de Sitter state. This thermodynamics includes also the gravitational coupling $K$ and the scalar Riemann curvature ${\cal R}$ as the thermodynamically conjugate variables. These variables modify the thermodynamics of the Gibbs-Duhem relation in the de Sitter state. The free energy density is proportional to $-T^2$, which is similar to that in the non-relativistic Fermi liquids and in relativistic matter with equation of state $w=1$. The local entropy is proportional to the local temperature, while the total entropy inside the cosmological horizon is $A/4G$, where $A$ is the area of the horizon. This entropy is usually interpreted as the entropy of the cosmological horizon. We also consider the possible application of the de Sitter thermodynamics to the Schwarzschild-de Sitter black hole and to black and white holes with the de Sitter cores.