  We discuss and expand a new approach to the thermodynamics of scalar-tensor
gravity and its diffusion toward general relativity (seen as an equilibrium
state) proposed in a previous Letter [Phys. Rev. D 103, L121501 (2021)], upon
which we build. We describe scalar-tensor gravity as an effective dissipative
fluid and apply Eckart's first order thermodynamics to it, obtaining explicitly
effective quantities such as heat flux, "temperature of gravity", viscosities,
entropy density, plus an equation describing the "diffusion" to Einstein
gravity. These quantities, still missing in the usual thermodynamics of
spacetime, are obtained with minimal assumptions. Furthermore, we examine
certain exact solutions of scalar-tensor gravity to test the proposed formalism
and gain some physical insight on the "approach to equilibrium" for this class
of theories.
