  Using the framework of stochastic thermodynamics we study heat production
related to the stochastic motion of a particle driven by repulsive, nonlinear,
time-delayed feedback. Recently it has been shown that this type of feedback
can lead to persistent motion above a threshold in parameter space [Physical
Review E 107, 024611 (2023)]. Here we investigate, numerically and by
analytical methods, the rate of heat production in the different regimes around
the threshold to persistent motion. We find a nonzero average heat production
rate, $\langle \dot{q}\rangle$, already below the threshold, indicating the
nonequilibrium character of the system even at small feedback. In this regime,
we compare to analytical results for a corresponding linearized delayed system
and a small-delay approximation which provides a reasonable description of
$\langle \dot{q}\rangle$ at small repulsion (or delay time). Beyond the
threshold, the rate of heat production is much larger and shows a maximum as
function of the delay time. In this regime, $\langle \dot{q}\rangle$ can be
approximated by that of a system subject to a constant force stemming from the
long-time velocity in the deterministic limit. The distribution of dissipated
heat, however, is non-Gaussian, contrary to the constant-force case.
