For museums, would it be an option to instead of a cooler have a temperature control unit that keeps the package at a set temperature no matter wether the PC is operating or not? Just heating the chips surfaces might be cheaper than having the full PC on 24/7 with a semi constant load.
The PC would still heat up when started. You would need very precise temperature control to avoid that. That could be quite difficult to do
With a couple of kelvins of tolerance a PID controller could handle this fine