My secondary air pump works I hot wired it but it does not come on when cold start

Have you checked the secondary air injection pump relay? The secondary air injection solenoid relay? These two are essential for proper system powering. The engine's processor grounds these relays to start the pump. I wouldn't bother looking into check valves or any of that because that comes after the pump functions in the first place. Good luck. Here's a link to shops near you for additional hands on assistance: www.repairpal.com/repair-shops