On my C230 the air pump does a self-diagnostic check once a day in the morning and it fits everything you describe although it occurs within the first 10 minutes. After driving about 5 to 10 minutes, when I come to a stop light it turns on for about 2 seconds. The engine stumbles briefly at this point and I can faintly hear a vaccum cleaner sound. It worried me at first until I learned what it was doing. Still seems kind of odd. I don't know if your car has that system though. The other ideas below look interesting also.

Mike

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1998 C230 330,000 miles (currently dead of second failed EIS, yours will fail too, turning you into the dealer's personal human cash machine)
1988 F150 144,000 miles (leaks all the colors of the rainbow)
Previous stars: 1981 Brava 210,000 miles, 1978 128 150,000 miles, 1977 B200 Van 175,000 miles, 1972 Vega (great, if rusty, car), 1972 Celica, 1986.5 Supra