Interesting problem… how do you store energy when it comes in the form of small heat differentials? That sounds like one hell of an interesting battery.

Maybe if there was also a natural source of cold, as well as a natural source of heat. The problem might then be to store the hot/cold inside a material until they can be mixed. Then mixing in such a way that allows energy extraction?

Except you wouldn’t mix them directly. You’d instead let them try to mix thermally, while forcing the heat to travel through something that allows energy extraction.