Cooling is rarely done in any other way than compressing a gas, allowing the heat to dissipate and then allowing it to decompress again. You don't want to compress a gas to cool another gas about to be compressed. What reasonably advanced compressed gas storage systems do is capture and store the heat that gets created during compression and feed it back during decompression. This gives the same efficiency difference as compressing some magically pre-cooled gas would do, only on the discharge side.

So far so good, just the old thermodynamics. It gets interesting when you have a cooling use case anyways: then you can skip on some of the decompression recovery and use the "cold" from decompression directly, to cool down something that needs cooling, without going the extra way of converting back to electricity and then sending the electricity recovered into a compressor setup for "creating cold". Bonus points if you also have a use case for the heat you did not use in reconversion to electricity, but chances are between losses during storage and heating some the gas back some amount beyond neutral you won't have much spare heat anyways.