A B757 crew refused an aircraft because the APU failed the high altitude inflight start verification but operated normally on the the ground. The weather at the next destination was marginal and the Captain wanted the APU operable.

Narrative

We brought this aircraft in and enroute on that first flight we were required to do an APU high altitude start verification. The APU failed to start. Upon arrival the APU started on the ground. Mechanics met the plane and started working on the problem. As push time approached we were told that boarding would be delayed because of the status of the APU. I proceeded to the ramp where several line mechanics and their supervisor were discussing the APU. I queried them on our status. The Supervisor then told the mechanics to stop working on the airplane and told me the Captain would have to refuse the aircraft for work to continue. I informed the Captain of this and he proceeds to call Dispatch and Maintenance. The Captain was informed by Maintenance that he would have to refuse the aircraft in order for work to continue. He refused the aircraft. The local mechanics replaced a part on the APU they thought was the culprit and the APU started and ran normally. We were again asked to do a high altitude start enroute and the APU failed all 3 start attempts. In this scenario the Captain felt the APU was necessary for safe completion of the flight based on the potential for marginal weather in our next destination.