  We investigate the occurrence of heat leakages in overdamped 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.
