05 Tacoma new air pump does not come on. No codes available pump runs when jump at relay

Normally I hear air pump go on when shut down??

 

Assuming you mean Secondary Air Pump it should run about 30-60 seconds on a cold start, I think only 13+ runs a few seconds at shutdown. I'm not sure if 4cyl trucks run it at shutdown or not, I'm assuming you have a 4 cyl since the V6 only had Secondary Air Injection on 2012 and up.

 

Yes 05 Taco 2.7
So it’s normal??

 

If no codes related to the Secondary Air Injection System are pending or stored I would have to assume so.
@Jimmyh has a 4cyl although I don't remember what year it is, maybe he knows if it runs at shutdown or not.

 

I believe that it does...

Anyway No codes = No worries.

When I drive my truck again I'll check and see if it runs on shutdown for sure.

 

Update:

Edit:

I drove around and ran some errands. Truck fully warmed up and the pump Did NOT Run when i shut the engine off.

 

No codes is your pump working
??