OBD 2 constantly runs monitors under all conditions. The non AC might be for the fast track all monitor test.

The secondary air test on my 97 runs after the engine is warmed up and returns to idle. It runs the air pump then looks for the O2 sensor to read lean ( low voltage for a traditional unit , for a wide band it just looks for a lean condition ) Also, in normal operation the air pump only runs in the first minute after a cold start.

I'd expect the 04 it use the same strategy, listen for the pump to run.