Can air have gravitational potential energy?
Yes. However, you often don’t notice this because as you lower a volume of air downward, you displace a similar volume of air upward. Thus you can’t just raise or lower air to observe changes in its gravitational potential energy. You’d have less trouble if you compressed the air tightly together, perhaps turning it into a liquid, and then raised or lowered it. It’s gravitational potential energy would then be much more noticeable.