Something simple you can try, and you probably already tried this but.......
I just got used 05 burb. Was looking throught the owners manual and thought I saw in there that if you have the temp controls off, then it will default to blow air at the temp it was set on before you turned them off. Mine was blowing cold in the rear also. Had my son turn the rear unit on and look at the temp, was set at about 70 degrees, so we turned the temp up and then it blew warm air wether or not the controls were actually "turned on" or not back there.
Have not ridden back there myself yet to really check it - so we are going on the word of a 12 yr old .
Mine is doing the same thing its a rear blend door accuator that is bad they are about $130 easy to change yourself just need to take off the pannels on the passenger side in the rear.i took mine out and just jerry rigged it to full hot for now until i get the $.I have good heat now just no way of controlling the temp back their except by just turning down the blower.Just have a auto parts store show u a pic of it or if u have the $ just get it and u will know what it looks like when u acces it after taking off the rear pannels.its held on by three small screws and has a quick disconect wire on it should come wright off.Then just install the new one,hardest part is takeing the pannels off.HOPE THIS HELPS.
c dailey is spot on…I had the same problem although both front and rear in my 2003 Suburban. Both actuators were bad. Approximately $150 each from local Chevy dealer, three hours total time for front and rear and I'm back to controlled climate! Just in time, temps in Chicago have finally started to drop. Just a side note, my front was blowing hot air on passenger side, cold on drivers side. The actuator that controls the distribution of warm air to drivers side is located on the passenger side under dash but closer to the center console (I learned there are a total of four upfront). Drop the black plastic cover 2-3 screws and its easily found tucked in underneath. Just be careful to NOT plug in the new actuator before installing it on the box. I was told powered up off the box potentially will allow the device to "travel" beyond the point of return. If adjustments are needed to line up the "keyed" slot turn the stem connected to the door inside the vent box. Extremely easy repair and would consider this level 2 (Beginner-Expert 1-5). Thanks for all the great advice!