the air pump in my car stays on for less than a minute. maybe a minute lately in the cold weather. i doubt that its not working has any real detrimental effect to the engine since its only purpose is in assisting in cleaning up the exhaust until the cat converter warms up sufficiently. it is not the same thing as the air pumps in older cars that had to stay on all the time...
1993 300e-2.8 - gone now <sigh>
"Do not adjust your mind, it's reality that's malfunctioning"