  We discuss the consequences of unique symmetry of de Sitter spacetime, which
is invariant under the modified translations, ${\bf r}\rightarrow {\bf r}
-e^{Ht}{\bf a}$, where $H$ is the Hubble parameter. Due to this symmetry, all
the comoving observers at any point of the de Sitter space perceive the de
Sitter environment as the thermal bath with temperature $T=H/\pi$, which is
twice larger than the Gibbons-Hawking temperature of the cosmological horizon.
This leads to the heat exchange between gravity and matter, and to instability
of de Sitter state towards the creation of matter, its further heating, and
finally to the decay of the de Sitter state. The temperature $T=H/\pi$
determines different processes in the de Sitter environment, which are not
possible in Minkowski vacuum, such as the process of ionization of an atom.
This temperature also determines the local entropy of the de Sitter vacuum
state, and this allows us 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
$K=1/(16\pi G)$. This supports the holographic properties of the cosmological
event horizon. 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. Such connection takes place only in the
$3+1$ spacetime, where there is the special symmetry due to which the variables
$K$ and ${\cal R}$ have the same dimensionality. We also consider the lessons
from the de Sitter symmetry for the thermodynamics of black and white holes.
