Compression heating/cooling

In desert climates, like the one where I was born, it's ridiculously hot in the day, and then very cold at night. People who live there spend a fortune on air conditioning, and gain a little more relief with swamp coolers, which work by evaporation. (Works great in dry desert climates!)
As a student of physics, I am aware that fluids (gasses and liquids) become higher in temperature when compressed, and lower when decompressed. I wish to harness this effect to replace air conditioning, and to shift costs to when energy is the cheapest, which is typically at night in deserts. (Day costs are high from high demand, everyone running their air conditioning at once.) If this system is widely deployed, then the energy pricing would be recalculated, and we'd have to switch to a battery system or something.
So we build under the house a huge air tank, compressor, and heat exchanger. At night, it's cold and energy costs are low, and we run the compressor. The compressed air in the tank gets very hot, and the heat exchanger blows the heat into the house. Aaaaaaah. Feels good. The tank is high pressure, room temperature air.
When the sun rises, the compressor is shut off. The tank under the house is now under huge pressure, likely many atmospheres worth. When the heat of the day starts, we open a valve and fill the house with a cold wind. (Because the same amount of heat to make a many atmosphere'd tank room temperature is very little heat at all when that air is reduced to one atmosphere's worth of pressure, as it would be when let out.) Aaaaaaaaahhhh. Feels really really good. The expander valve should only allow a small amount of air out for maximum results. The tank will also get very very cold as it does this.
By the time the sun goes down, ideally this is when our tank has reached one atmosphere of pressure, ending the cold wind. It's time to repeat the cycle, but we won't start compressing immediately. We should probably let it sit a few minutes. Then we compress it up.
I got this idea thinking about a museum I visited once that talks about how it shifted its cooling energy burden to nighttime, when energy demand is least, by freezing a whole lot of ice and using this ice for cooling during the day, when energy demand is higher. And I thought, why bother with ice?