High-tech Ents foresee a calamity befalling their beloved and treasured green planet. Surface temperatures will rise well above 500 Celsius (arbitrary, basically just very hot) for thousands of years, before cooling back down.
They enact many preservation plans, one going towards space, one going underground. Their rocket science is very limited, but their bioengineering is at near miraculous levels. For this reason they prefer to go underground, where they have less initial limitations. (All other tech, especially material manufacturing are high level too, since nature produces some amazing materials)
They have the capability to dig deep and create huge cavern habitats with different biomes to save much of their current wildlife. They already modified numerous organisms to manage water and waste recycling, humidity, inner temperature, oxygen (air), light, etc. and they are mostly autonomous, requiring no maintenance. They will however need constant energy and produce heat.
The underground habitats will be surrounded by extreme high temperatures from below (core heat) and above (the calamity), closed off from natural sunlight (free energy).
So the Ents face 2 big problems:
Even fully insulated from outside heat, all life produces heat. If they can't dump the excess heat, they will eventually cook themselves. Especially that their environment maintaining organisms produce even more than normal. Storing it in some form is not preferred for fear of the calamity lasting longer than calculated.
Even with vast stockpiles, they can't store enough energy inside these habitats to last forever. They will have to generate it some way using their current environment.
These are the two main problems the Ents identified so far, and can't solve without handwavium technology. The Ents, close to nature, abhor any form of handwavium. However in order to save the many lives of their world, they are prepared to put up with the least amount of it.
Is there a reasonable scientific way to solve the problems given the circumstances? If not, what type of handwavium should the Ents invest?
The solemn Ent researchers look towards the starry sky for answers.
Edit: though it isn't the main issue, the Ents envisioned their habitat's "shell" as layers of high melting point, low heat exchanger material with close to vacuum conditions in between. They still looking into the best composites, but figured if they can dump the inner heat outside, they could do the same with any leaked in heat.
Edit 2: The "miraculous" bioengineering is their own, arrogant estimate. Basically, they can produce machines and infrastructure which take care of its own maintenance, given steady energy supply. It also helps with the long-term self-sufficient biome designs, given the 2 connected problems solved.
Edit 3: Ents - tree like plantoids, sentient, biological. Technology, culture somewhat similar to humans but more heavily emphasize co-existence and symbiosis with nature.