Einstein gravity as the thermal equilibrium state of a nonminimally coupled scalar field geometry

We test ideas of the recently proposed first-order thermodynamics of scalar-tensor gravity using an exact geometry sourced by a conformally coupled scalar field. We report a non-monotonic behaviour of the effective ``temperature of gravity'' not observed before and due to a new term in the equation describing the relaxation of gravity toward its state of equilibrium, i.e., Einstein gravity, showing a richer range of thermal behaviours of modified gravity than previously thought.