  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.
