Yes, but you need energy to pump heat, and that has an efficiency maximum (thx ~~Obama~~ Carnot), and radiative cooling scales with the ~4th power of the temperature, so it has to be really hot, and so it requires a lot of energy to "cool down" the already relatively cool side and use that "heat" to heat up the other side that's a thousand degree hotter.

All in all, the cooling system would likely consume more energy than the compute parts.