Classic 737's apu's will only run to 18,000ft then they cough!
- not in my experience. They will run far higher than that.
As an 'aside', in the 'old days' with DanAir on the LGW-CDG hops (737 3/400, and cruise normally at 250) we found that leaving it running (on the manufacturer's recommendation) drastically increased the MTBF and maintenance costs due to the reduction in the short-cycle start-up/shut-down thermal shocks and more than paid for the minimal fuel burn.