Phase change isn't a necessity it's a change in pressure. The air in the tank and airline is hot but when discharged through an orifice the pressure drops and the air will absorb heat as the pressure has dropped.
Refrigeration works the same way but for it to be efficient you also need the phase change as the most heat is absorbed or consumed depending on the situation.
Now to use an air compressor for heating or cooling yes compressed air is used to do that in industrial applications but it's extremely inefficient and noisy.
You can get a cold gun which uses shop air to create cold air and hot air as well. It's voodoo magic in a tube.
https://www.exair.com/products/cold-...t-systems.html
In short air compressors are the biggest energy hogs in the shop. They are noisy produce allot of heat and use allot of electricity. And I need a much larger one for my shop as my customized 60 gallon doesn't keep up with my needs anymore.
Couldn't imagine what it would take solar panel and inverter wise to run an air compressor...