Laws of thermodynamic equilibrium within first order relativistic hydrodynamics

Using recently developed consistent and robust first order relativistic hydrodynamics of a dissipative fluid we propose a generalization but weak version of Tolman-Ehrenfest relation and Klein's law on a general background spacetime. These relations are appeared to be a consequence of thermal equilibrium state of the fluid, defined by the absence of heat flux. We interpret them as the defining relations for the local temperature and chemical potential of the fluid. The validity of usual Tolman-Ehrenfest relation and Klein's law deeply depends on the existence of a global timelike Killing vector. However imposition of more stronger equilibrium condition -- local conservation of entropy current -- yields the constancy of the equilibrium thermodynamic parameters along the flow lines.