I had a similar question, but mine was if the cave failed (unable to keep it cold enough through summer) and the cheese needed to be placed in a normal fridge, would it ruin the cheese.
I think the general thought was that it would be fine, it would just mature at a slower rate. So that would be my thoughts here.
My only question about adding a heater, would be drying the air too much, and causing cheeses to crack. Just a thought.