Electric?
The whole purpose of APUs is to generate electricity independently of ground aids. Air bleeds to give conditioned air is secondary. Thus, using an APU that is fuelled seperately would require a seperate fuel tank, adding weight and complexity.
So, basically, if APUs didn't run from the main fuel source, they wouldn't be worth considering. If you look at an APU, and then at a GPu unit, you'll see the logic. (give or take the trailer chassis attached!)