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Using recently developed consistent and robust first order relativistichydrodynamics of a dissipative fluid we propose a generalization but weakversion of Tolman-Ehrenfest relation and Klein's law on a general backgroundspacetime. These relations are appeared to be a consequence of thermalequilibrium state of the fluid, defined by the absence of heat flux. Weinterpret them as the defining relations for the local temperature and chemicalpotential of the fluid. The validity of usual Tolman-Ehrenfest relation andKlein's law deeply depends on the existence of a global timelike Killingvector. However imposition of more stronger equilibrium condition -- localconservation of entropy current -- yields the constancy of the equilibriumthermodynamic parameters along the flow lines.

Laws of thermodynamic equilibrium within first order relativistic hydrodynamics