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The Tolman-Ehrenfest criterion for the thermal equilibrium of a fluid at rest in a static general-relativistic geometry is generalized to scalar-tensor gravity. Surprisingly, the gravitational scalar field, which fixes the strength of the effective gravitational coupling, does not play a role in determining thermal equilibrium. As a result, heat does not sink more in a gravitational field where gravity is stronger., Comment: Conclusions changed, explanation and bibliography expanded. Matches version to appear in EPJC

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., Comment: 13 pages, latex, one figure