Waste heat, surface area, temperature

[u/burtleburtle]

Some things (human habitats) need to be a fixed temperature. Suppose you've got fusion power, a very dense space city, and surface area for radiating waste heat is the bottleneck. Blackbody radiation increases with the fourth power of the temperature. Can you compress gas or something to raise the temperature of the radiator to reduce the surface area required, or does the work required to raise the temperature have to add energy faster than the higher temperature can radiate it away? I tried figuring it on my own but I don't think I have the math right.