  Based on the recent work [1,2], we formulate the first law and the second law
of stochastic thermodynamics in the framework of general relativity. These laws
are established for a charged Brownian particle moving in a heat reservoir and
subjecting to an external electromagnetic field in generic stationary spacetime
background, and in order to maintain general covariance, they are presented
respectively in terms of the divergences of the energy current and the entropy
density current. The stability of the equilibrium state is also analyzed.
