That makes radiating a much more practical approach to cooling it.
Yes, running hotter will cause more energy to be radiated.
but
These parts are not at all designed to radiate heat - just look at the surface area of the package with respect to the amount of power they consume.
Hmm, surely the radiator can run at arbitrary temperatures w.r.t. the objects being cooled? I'm assuming heat pump etc is already part of the design.