We investigate the occurrence of heat leakages in overdamped nonequilibrium Brownian harmonic systems. We exactly compute the underdamped and overdamped stochastic heats exchanged with the bath for a sudden frequency or temperature switch. We show that the underdamped heat reduces to the corresponding overdamped expression in the limit of large friction for the isothermal process. However, we establish that this is not the case for the isochoric transformation. We microscopically derive the additionally generated heat leakage and relate its origin to the initial relaxation of the velocity of the system. Our results highlight the limitations of the overdamped approximation for the evaluation of the stochastic heat in systems with changing bath temperature.