No we haven't been paying so it's costing them each time they come out. It's like a bad joke.
So it's not like they are scamming you, just that they are incompetent to fix your problem.
I've had this happen with auto repairs before. Same guys tried to fix the problem 6 times and the problem kept returning within a week of being in the shop. In my situation, they were really trying their best and were willing to pay for me to have a rental in the meantime, and honestly trying really hard to make things right. So I let them continue to try to fix the problem. Ultimately they couldn't fix it and refunded me my initial payment so that I could take the car to a specialist (transmission shop).
If I were you, I'd expect the same. It sounds like you've given them appropriate time to fix the problem and they are unable to do so. If your air still does not work, then they haven't held up their end of the contract. They should refund the money or else you can sue them. This should be a no brainer in small claims court.
Hey Judge, I paid 4500 bucks for company X to fix my A/C. My A/C is still not fixed weeks later. I then had to go to company Y to fix the problem. Company X should refund my payment because they did not fix the problem according to our original agreement. done and done.
Edit: the downside here is obviously that you'll have to wait a good while for your day in court. In the meantime you'll have to fork out more $$$ to have another company fix your problem. However, keep all documents from both companies...bills, receipts, invoices detailing work performed, etc. That way the judge can see just what happened and what was done.