If you're operating near a sun ("near" being a relative term, but inside 2 AU is a quick rough estimate), you will probably need to have good heat management to deal with all the energy hitting the ship's skin from the sun. Even if you're far away from the sun, you will still need some heat management to deal with the heat generated by your ship's engines, equipment, and personnel. In either case, you have to deal with moving heat around.
Stirling engines generate power from the movement of heat from one place to another. They do so with higher efficiency and using lower gradients than many traditional engines. You can also put power into a Stirling engine to move heat against a gradient.
So, would a Stirling engine interact well with a heat management system? Unlikely but it's plausible.
If I wanted a pseudo-realistic version of this, I would go for something like an outer hull layer with many radiative/absorbtive baffles. That lets you pull heat in or radiate it out as efficiently as possible (and it might look cool. 184
Ships could have fur, or moss, or whatever other motif). Inside that, wrap the ship in a layer of Stirling pumps, all in battery. You'll get something like a big Peltier-effect blanket. It would take a little handwaving to say why this is better than just using Peltier elements, but that's just an engineering problem ("the shared gas pool does better with load-balancing for hot spots; the mechanical system is more resilient to cosmic rays; the medium provides additional shielding; Peltier belts tend to short out in big chunks; etc).
So, that gives you an element of your ship which protects from heat, provides warming and cooling, and generates some power (maybe) near anything that pumps heat into your system. If stealth is important to you, this might provide that too (pump all the heat out of the side of the ship facing the sensor). Functionally, you have something that can degrade without breaking, can leak and be fixed with chewing-gum-style patches, makes noises as needed, is very important to the safe operation of the ship, and isn't immediately fatal on failure. Also, it can run on whatever unobtanium you want as its gas medium. 898