I have had the same trouble. Usually when the exhaust has reached temperature the secondary air injection pump switches off. I assume you also get exhaust gas smells when you turn on the heater/ventilation fan. If this is the case you may have what happened to my Vectra. The secondary air injection valve failed allowing hot exhaust gases to go back into the pump effectively writing it off. Rectification is simple. I obtained a seconhand secondary air injection pump from a well known auction site and a new valve from the vauxhall dealer. the valve took minutes to replace and the pump can be replaced by:
1. Remove the battery by removing the securing clamp.
2 Undo the three bolts (13mm socket) which hold the mounting plate in place.
3 Unclip the electrical plug to the pump.
4 Unclip the air tube from the pump.
The pump should now be free. Check that the mounting plate and rubber mountings are the same as the ones you have on the original.
Re-fitting is the reverse of removal and a reasonably competent home mechanic should do it in half an hour.
The failure to start probably has nothing to do with the secondary air injection system, more likely to be the battery.
Hope this helps.
Steve Sidney, September 2011