We derive the master equation that completely determines the time evolution of the density matrix of the Unruh-DeWitt detector in an arbitrary background geometry. We apply the equation to reveal a nonequilibrium thermodynamic character of de Sitter space. This generalizes an earlier study on the thermodynamic property of the Bunch-Davies vacuum that an Unruh-DeWitt detector staying in the Poincaré patch and interacting with a scalar field in the Bunch-Davies vacuum behaves as if it is in a thermal bath of finite temperature. In this paper, instead of the Bunch-Davies vacuum, we consider a class of initial states of scalar field, for which the detector behaves as if it is in a medium that is not in thermodynamic equilibrium and that undergoes a relaxation to the equilibrium corresponding to the Bunch-Davies vacuum. We give a prescription for calculating the relaxation times of the nonequilibrium processes. We particularly show that, when the initial state of the scalar field is the instantaneous ground state at a finite past, the relaxation time is always given by a universal value of half the curvature radius of de Sitter space. We expect that the relaxation time gives a nonequilibrium thermodynamic quantity intrinsic to de Sitter space.