If you recharge your a/c system with freon (bottled goodies from local parts store) and notice the cold goes away, then you have a leak. Your A/C system is sealed from factory. I would get a bottle that has a dye additive and stick a flourescent bulb 2-3" away from all of your system and check for leaks. It will shine where the leak is. Ensure you run from the compressor through the high pressure side-H Block (TXV)-low pressure side-muffler (there is a muffler on your a/c system believe it or not), back to the compressor again.
If you have a friend around, pop your hood and start the car. Have him monitor where the compressor is and tell each other when a/c is "on" and "off". See if he notices the tri shaped object inside the pulley moves or not. That can determine easily if your compressor is the cause too. Our cobalts use a modern wobble plate type.
2009 Cobalt LS XFE U74
Mods: K&N Typhoon Intake, Option Cat-Back Exhaust, Cosmo Short shifter, Maxxim Ahead 17X7's, MPx shorty antenna, FE5 Strut/shocks, Tein S Techs, Moog endlinks, Moog spring seaters, ZZP shorty header, LSJ downpipe, LNF front grille