Have you ruled-out the blend door being stuck?
I don't know that the corolla employs this particular part, but I had this happen on another vehicle. There's a mechanical piece that physically blocks off the hot/cold air tubes. It's a simple thing to fix, but a lot of labor to get to it. I actually sold my Ranger rather than paying $700+ to have someone take apart the dash and fix it. The ranger had 160k+ miles, and it got crap mileage, so it wasn't a hard decision for me.
Your car is obviously a lot newer, but you may want to search and see if that's a problem others have encountered.