  We present results for the $gg$-part of the one-loop corrections to the
recently introduced "topological charge" GPD $\widetilde F(x,q^2)$. In
particular, we give expression for its evolution kernel. To enforce strict
compliance with the gauge invariance requirements, we have used on-shell states
for external gluons, and have obtained identical results both in Feynman and
light-cone gauges. No "zero mode" $\delta (x)$ terms were found for the twist-4
gluon GPD $\widetilde F(x,q^2)$.
