Nothing drastic happens when the air pump fails. It is designed to come on far a minute or two after a cold start to reduce cold start emmissions. It is fairly obvious and definitive noise that comes on and goes off when its time is up. It will not come on if the engine is started warm or hot.

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Ali Al-Chalabi

2001 CLK55
1999 Dodge Ram 2500 Cummins Diesel
2002 Harley-Davidson Fatboy
Merlin Extralight w/ Campy Record