THe criteria that determine when the ECU will run the pump is not clear to me. I don't always hear the pump on startup.
You can google "Secondary AIr Injection" and learn WHY it runs and what the criteria are, but like me, you will probably find conflicting information.

My understanding:
The pump is run in order to add air (which means you add more oxygen) to the exhaust gasses, this helps cause a reaction in the cats that helps them heat up faster (they do their job better when hot).
The ECU 'knows" if the pump runs and does it's job by seeing an increase in O2 level via oxygen sensors in the exhaust.
If the ECU 'gets the idea' that the cats are already probably hot, it won't run the pump - but I don't know how it comes to that conclusion, other than monitoring water temperature, or maybe it has a memory of how long it's been since the engine last ran or maybe if the ambient or intake air temp is over a certain temp. Hard to say, but I do know that my pump does not always run on startup even with a cold engine.