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

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