Thermodynamics of scalar-tensor gravity

Previously, the Einstein equation has been described as an of state, general relativity equilibrium state gravity, and $f({\cal R})$ gravity a non-equilibrium one. We apply Eckart's first order thermodynamics to effective dissipative fluid describing scalar-tensor gravity. Surprisingly, we obtain simple expressions for heat flux, "temperature gravity", shear bulk viscosity, entropy density, plus generalized Fourier law in consistent Eckart thermodynamical picture. Well-defined notions temperature approach equilibrium, missing current spacetime scenarios, naturally emerge.