I think this will work, but only on a rather small scale. To cool down the air in an entire house is going to require a very large volume of compressed air. As you point out, the heat has to be removed from the air after it is compressed, so it has to be stored in a pressure vessel while the heat is removed.
You can get some idea of the volume required if you think about simply mixing the expanding air with the air that is already in the house. It's going to need a lot more than you can get even from quite a large air compressor.
One way around this would be to actually liquefy the air. The liquid air will take in a lot of heat when it returns to a gas. However, it's not so easy to liquefy air. The process requires refrigeration and compression.
That is why conventional AC systems use refrigerants like CFC. When the refrigerant changes state from liquid to gas, it takes in a huge amount of heat, so a relatively small volume of refrigerant can transfer a lot of heat energy.
It struck me there is another problem with the compressed air approach. The energy used to compress the air does two things. It heats the air up and it also compresses it (strangely enough!). The heat energy is dissipated, but some of the energy is stored in the cooled compressed air. (It's analogous to winding up a spring.)
When the compressed air is allowed to expand it does absorb heat, but the energy stored in the compressed air is doing no useful work, so that energy might simply be wasted. However, I suppose you could do something useful with it like assisting the compressor. The expanding air might also do work in a Venturi which can produce some very low temperatures.