neat. why can't you put the heat in the mountain?

For broadly the same reason that many years of heat from braking has heat-saturated the london underground.

https://www.google.com/search?client=firefox-b-d&q=london+un...

You could probably drill sideways into rocks and pump heat into the mountain for a certain period of time, but on a multi-month or year time scale, it would catch up with you after you had saturated it.