I was unfortunate enough to have flown classic B737-400 into places without ground power to support the aircraft for a decade some 20years ago. Coupled with ‘not-so-sharp’ grease monkeys at home base workshop, APU start success( on the ground) was a an issue. Low engine bleed pressure, sticky APU start valve were some excuses those grease monkeys mentioned. I know nothing because I am just a pilot. Carrying extra fuel for one engine running transit would put landing performance beyond the curve on the 5600ft runway.
Starting the APU on descend successfully put some worries to rest. No scientific proof but it seems APU has better start success in the descend compare to on the ground.
I would not recommend this practice to anyone in this era where everyone is waiting to shoot you bum-bum if you deviate from the books.