That heat isn’t the geothermal (i.e. picked up from the walls of the deep cave) heat though. That’s heat that was generated by the compressor when it compressed the air (so the energy source is the electrical input to the compressor - it’s basically waste heat). But that captured “waste” heat needs to be put back into the air when it’s decompressed, otherwise the temperature would drop far too low when it’s decompressed which causes an efficiency drop at best, and damage to the turbine at worst. Other compressed air storage systems use burning gas to heat up the air while it’s being decompressed, but this new design reuses the heat generated at the input for improved efficiency and no dependence on fossil fuels.

So the water tanks aren’t related to my thought about picking up extra incidental geothermal heat.