We start with one small (but perfectly formed) planetoid. Just for the sake of discussion, we'll say it's Mercury. This planet is more or less useless to me, being too small and too close to the sun. Fortunately, I've got this big building project in mind...and I need materials.
Not-Mercury is therefore slated for deconstruction. I have massive mobile excavator/refinery complexes that are eating steadily away at the crust, processing the rock into useful components, and transporting them to orbit for use in my Dyson Swarm.
However, the core of the planet is still extremely hot - not as hot as Earth's core, and not hot enough to drive tectonic activity, but certainly hot enough to cause critical components of my excavators to become inconveniently runny. What I'd like to do, then, is have some process in place that can transport the heat away from the depths of the planet, keeping the surface cool enough for my machines to work efficiently.
The population of the planet live in subterranean, multi-level tunnel-cities (similar to Zion in the Matrix series). Ideally these cities should be a part of whatever system we put in place to cool the core, but that's not strictly required. Tech level is advanced enough to allow routine interplanetary travel, but not interstellar; fusion reactors, mass drivers, arcologies, etc, but no FTL, matter replication, or transporters. The system is almost completely bare of Handwavium, so ideally tech should be conceivable today.
So - How can I cool down a planetary core on a timescale of centuries?