There are many airports between 5000 and 10000 with 737s starting their APUs on them.
Should not have any associated issues. Depending on the specific type of APU certain tweaks may be made to provide for quicker acceleration.
I do not believe that to be the issue. Thinner air makes it harder to start. I believe the advantage to be a cold soaked APU and 5K being a good altitude to balance fuel burn v/s a nice cold APU.