The 150-year-old paradox of Maxwell’s demon is shown to be resolved by information thermodynamics that describes information-scientific notions, memory, information, and computation, from thermodynamic theory. Memory is seen as a thermodynamic state of a physical device or demon, information acquisition as the gain of entropy from a measurement process, and computation as a post-measurement process by feedback of the measurement outcome. We investigate a general feedback control with multiple steps of measurement/feedback and generalize to a case for the measurement of an observable with odd parity in time reversal such as momentum. We examine the effect of a time delay and a finite-time duration in appying a feedback protocol which are unavoidable in real experiments. We exemplify a cold damping problem where a velocity of a particle is measured and a dissipative protocol is applied by feedback. We show how the effective temperature of the system is updated (cooled) towards a steady-state value through repeated feedback processes. We find the existence of the instability region for steady-state temperature in the parameter space of the time duration, the time delay, and the dissipation coefficient for feedback protocol.