Cosmological constant as a source of Hawking entropy in de Sitter space

The de Sitter space is a solution of the Einstein equation with a cosmological constant. In this space, there is a horizon with which the Hawking entropy is associated, and an observer in the de Sitter space sees the temperature. Assuming that the contribution of the cosmological constant to the vacuum energy depends on the temperature, we will show that the entropy of the entire space bounded by the horizon coincides with the Hawking entropy. Thus, in the de Sitter space, we can talk about the presence of local thermodynamics: the de Sitter vacuum can be considered as a thermal state, and it serves as a source of the Hawking entropy.
We will also generalize this result to the theory of f(R)-gravity, which has de Sitter solutions and a cosmological constant.