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Heat always flows from hotter to a colder temperature until thermalequilibrium be finally restored in agreement with the usual (zeroth, first andsecond) laws of thermodynamics. However, Tolman and Ehrenfest demonstrated thatthe relation between inertia and weight uniting all forms of energy in theframework of general relativity implies that the standard equilibrium conditionis violated in order to maintain the validity of the first and second law ofthermodynamics. Here we demonstrate that the thermal equilibrium condition fora static self-gravitating fluid, besides being violated, is also heavilydependent on the underlying spacetime geometry (whether Riemannian ornon-Riemannian). As a particular example, a new equilibrium condition isdeduced for a large class of Weyl and f(R) type gravity theories. Such resultssuggest that experiments based on the foundations of the heat theory (thermalsector) may also be used for confronting gravity theories and prospect theintrinsic geometric nature of the spacetime structure.

Tolman-Ehrenfest-Klein Law in non-Riemannian geometries