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Sounds like the complete inverse of the cooler.

How would you power such a thing?  I'd imagine you'd need wood to burn (a campfire) at one end and ice to freeze (or at least a passive cooler) on the other to make it work... and at that point, using the wood-fired generator would be a less-expensive power source.

I think it would look more like vent that get electricity from equalizing temperatures.
We would need something to burn. It is the point. You could haul corpses, clothing, hay, anything that burns in a double-walled room, then spark it and have energy. When there will be no fuel, it can generate ~50W from daily temperature change.

In order to make a lot of energy through the stirling process, one side needs to be extremely cold, while the other side needs to be extremely hot.  This kind of thermal difference is not possible in the real world without first expending energy to heat one side, expending energy to freeze the other side, and provide sufficient insulation between the two to keep those respective sides hot and cold.

Like I said, it would be way more expensive to get a stirling engine to the same power generation as the wood-fired generator.

In wood-fired generator, you can burn only wood. If you use sterling, you can burn anything.

A structure that generates electricity from difference of temperatures, equalizing them in process. It would be great to build clothing generators, skeleton generators or even daily temperature change power plants, with it we could get energy from everything.

Maybe implement "Rogue Planet" event?

Temperatures always drop, -2 degrees per season. Possibly add bottom, where temperatures don't go lower, close to -100 (you can't cool anything down -270 already) to make lategame survivable. Rename it to "tidal lock" or "dark planet".