Your issue could be more complex than this but it could also be part of the problem.
http://www.hummer4x4offroad.com/forum/showthread.php?3962-Climate-Control-fix
http://www.hummer4x4offroad.com/forum/showthread.php?3962-Climate-Control-fix
Okay soo after testing I've found that if I let the car sit for an hour. Come back and start it up the air blows cold for about 20 mins then the compressor starts kicking on and off on and off. I'm thinking low freon, so the compressor turns off to save it self then the freon psi builds back up and the compressor kicks on until the Psi drops again and it cuts off again? Thoughts?
I'm not sure if you fixed this or not, but I am having a similar experience. I put my gauge on and the pressure was reading low, so I added. The compressor started to cycle more, so I stopped and took it to a shop. I paid $73 to have them evacuate the system, vacuum it out, then recharge to the proper level (so it's definately the way to go). They said the system was slightly overcharged. When I got the truck back, it wouldn't blow cold, just slightly cool. I swapped out the relay and looked at the electrical connections and all were good. I decided to put my gauge back on it and it read 0. Long story short, before heading back to the shop to chew ass, the next day, I put my gauge on another vehicle and it read fine. I put them back on the H3 and it read low, like before, but would actually blow cold air.
Now, to the fun part. I discovered that our AC systems have a Thermal Expansion Valve (TXV). When the TXV starts going bad, it can cause the compressor to pull a vacuum on the low side or at least have a lower pressure that it should have. The low pressure causes the compressor to kick on and off and when it has a vacuum, it's not going to cool at all.
I have an appointment on Wednesday to get this part replaced. Will update once completed.