You will be fine without the AP. I don't know for sure about what Audio said in relation to the pump circulating while at idle speed, but I do know it is used when the car is off and you hit the MAX button and have heat while the car is off for a predetermined amount of time. Just fix it when you get a chance. No harm at all

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Dave S.
Northeast
2003 4.6is
2005 E53 4.4
2003 E39 Msport(a REAL ONE!)
1959 Fiat 600