  Lattice computations of the high-temperature topological susceptibility of
QCD receive lattice-spacing corrections and suffer from systematics arising
from the type and depth of gradient flow. We study the lattice spacing
corrections to $\chi_\mathrm{top}$ semi-analytically by exploring the behavior
of discretized Harrington-Shepard calorons under the action of different forms
of gradient flow. From our study we conclude that $N_\tau = 6$ is definitely
too small of a time extent to study the theory at temperatures of order
$4~T_\mathrm{c}$ and we explore how the amount of gradient flow influences the
continuum extrapolation.
