As distinct from the black hole physics, the de Sitter thermodynamics is not determined by the cosmological horizon, the effective temperature differs from the Hawking temperature. In particular, the atom in the de Sitter universe experiences thermal activation corresponding to the local temperature, which is twice larger than the Hawking temperature, $T_{rm loc}=2T_{rm Hawking}$. The same double Hawking temperature describes the decay of massive scalar field in the de Sitter universe. The reason, why the local temperature is exactly twice the Hawking temperature, follows from the geometry of the de Sitter spacetime. The weakening of the role of the cosmological horizon in de Sitter universe is confirmed by considering Hawking radiation. We discuss the difference between the radiation of particles in the de Sitter spacetime and the Schwinger pair creation in the electric field. We use the stationary Painleve-Gullstrand metric for the de Sitter spacetime, where the particles are created by Hawking radiation from the cosmological horizon, and time independent gauge for the electric field. In these stationary frames the Hamiltonians and the energy spectra of massive particles look rather similar. However, the final results are essentially different. In case of Schwinger pair production the number density of the created pairs grows with time, while in the de Sitter vacuum the number density of the created pairs is finite. The latter suggests that Hawking radiation from the cosmological horizon does not lead to instability of the de Sitter vacuum. The other mechanisms of instability are required for the dynamical solution of the cosmological constant problem. We consider the possible role of the local temperature $T_{rm loc}=2T_{rm H}$ in the decay of the de Sitter space-time due to the energy exchange between the vacuum energy and relativistic matter with this temperature.
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