We used to leave the APU running through the after takeoff flow and start it up descending through 10,000ft.
Then fuel got expensive.
Now it gets turned off at the first opportunity and started again at the last possible minute. To the point where we're cross-bleed starting second engines on the ground and coming to the gate and leave a single engine running until ground power can be hooked up.
I'm not sure anybody is doing the math on the fuel thrown through engines for cross-bleed starts and the cost of running an engine at the gate as opposed to the APU. I think it's based on simple airline management thinking of "APU bad" so we don't run it.