Hello there! The short answer is that geophysical cooling plays a role in shaping Earth's surface but not a whole lot, at least not yet.
The main theory for Luna (Moon) is that it was formed during a massive collision with Earth billions of years ago, and a chunk of Earth was lost and is now orbiting us (the Moon). The Moon may have had tectonics during the early parts of its life but has since seized, hence why cooling mechanisms make up most of the surface of the Moon.
However data on Mercury support active tectonics. Mercury is Earth-like regarding tectonics but has temperatures that drop to -290 F. That could be the culprit for the observed cooling surface structures on Mercury.
Earth is actually losing its heat very slowly as the tectonic engine wears out. Geologists predict Earth will almost duplicate what Mars is now in about ~2 Billion years, A cold rocky body with little to no tectonic activity. During that time geophysical cooling could play a larger role in shaping the Earth's surface.
