Check your alternator by disconnecting the power lead to the alternator and see if the power sucking/draing goes away. I have a story for you... By the number of reports I've seen just like this one, I'd say this is a very common problem on the Cobalt!
This is a situation that happened with my daughter's Cobalt just this week. She arrived home and the car reported on the readout that it was "entering battery saver mode" or something similar. I did some searching and found out that this means that the voltage has dropped below 10.7 volts. This was strange because the battery is fairly new and she had no prior issues. She also reported a strange "whining" noise from under the hood. When she went out to start it a couple hours later, it was DEAD, DEAD, DEAD... As in deader than I've ever seen any car that has just been run a couple hours earlier. I put my heavy duty battery charge on it and it pegged the amp meter at 70 amps. This was truly a dead short situation. I've never seen anything like it. I charged the battery overnight and got back on it the next day to see if it would start, which it did. The whining noise under the hood was pretty easy to hear and the battery was being sucked dry again. I put a fresh battery in it from a different vehicle and again, dead short, heavy draw from the battery terminals. Huh... That's really weird. I started it again and I could smell something "plastic like" burninng near the alternator. I stuck my nose down there near the alternator and i could smell it even more and I could feel the heat coming off the alternator as well. The alternator was smoking hot... I couldn't even touch it. This was after just a few minutes of running. I used my lifeline... Called a friend who is a Pro mechanic and ran this scenario by him. His answer: Duh... Your alternator is toast! If you unhook the power cable from it, your problem will be solve immediately. He was absolutely right.
I've been working on cars all my life. I've never heard of an alternator doing this - Dead short, killing the battery and everything that goes with it. The car was completely dead and it could not be brought back with a 100 amp charger. Even with the charger on high, the charger could not overcome the shorting out of the alternator.
I've seen this same set of symptoms posted on Cobalt forums all over th place with no one seeming to come up with a solution. I hope this helps another Cobalt owner. The fix is pretty easy (test it first by disconnecting it though) Swapping out the alternator is about a 20 minute job and the part is $150-$350 depending on whether or not you go with a rebuilt or a new one. Rockauto has a new AC Delco for about $230 with no core charge. Not cheap but not horrible either.