11 hours ago, Jeeper said:

But 3 times since (and without the wine) I left the APU on until cruise to see if I was imagining things and, lo and behold, it happened all three times. 

As noted above, this makes sense because when you turn off the APU, the engines now need to supply the bleed air. This reduces thrust and hence airspeed.

As a test, you could turn only the APU bleed air switch off, without turning the APU itself off. You should see the same reduction in airspeed.

5 hours ago, martinboehme said:

this makes sense because when you turn off the APU, the engines now need to supply the bleed air.

However - during cruise power setting the engine bleed air should have a higher pressure than the APU bleed air and check valves would close to avoid back pressure flowing into the APU...so it should not have any effect - certainly not enough for the pilot to notice the change in bleed air extraction. You can test this by turning off the engine bleed valves and see if this "increases" airspeed.

Besides, changing bleed air extraction would only increase the EGT (with the N1 remaining constant as set by the autothrust), as the FADEC would just command a higher fuel flow to keep the N1 constant. The engines at cruise do not run at the EGT or RPM or pressure limit, so extra bleed air extraction can be compensated simply by burning more fuel.

Caveat: I have only flown A319/320/321 aircraft for ten years and the last time was 4 years ago.