I don't think it will matter much though. Whether you have to remove input power as heat or expend power to cool likely won't make a whole lot of difference. Just that the delta-t will be a lot larger and so it may actually cost more to keep that ultra-cool machine ultra-cool.

Maybe future data-centers will be used to pre-heat the water that goes into the turbines that power the electricity plant, in some kind of perverse version of a perpetuum mobile.

You'd need to make up for the difference lost on each cycle with extra fuel.

Or we could put future datacenters in space behind sunshades and get them really cold by default.

You need really big radiators for that. In space, waste heat is a huge problem.

Bandwidth might leave something to be desired.