Wyoming Wind and Utah Storage Light Up LA

A lot of hot air that cools down when stored and very poor energy efficiency.

Sounds like Washington, D.C., alright.

Yes, the secret to efficient energy storage with compressed air is to conserve heat. That's straight forward engineering. Use a huge pressure vessel, or insulate it, or use phase change materials to reduce temperature gradients. For example, if the hot compressed air is used to make steam, and the steam is stored in insulated vessels, the steam can be used to reheat the air before it is expanded, improving overall efficientcy.

www.isentropic.co.uk has the correct answer, and does not require underground caverns, but they do run their compressors hot for a reason, and have pressure vessels at some interestingly high temperatures, and one vessel with parts at a really cold temperature.