B757 Captain reports APU fault light after second engine start at gate; and agrees to continue flight with deferral and repairs at next stop. Reporter is contacted by duty manager as to reason for refusal at next stop.

Narrative

On day 2 of our 4 day trip; we pushed off the gate. After a normal start of both engines; I looked up to turn off the APU to find a fault light and the APU flamed out. We contacted maintenance and taxied out to the active runway. We discussed the APU inoperative and agreed to continue flight to LAX only; with a deferral. I advised maintenance we needed the APU fixed prior to the next leg. All parties were in agreement; pilots; maintenance and Dispatch. XA40L; Duty Manager called me on my cell. The reason for his call as he stated was to get an explanation of why I refused the aircraft. He apologized for having to ask this question; but stated they were now required to do so. He also suggested that I keep all pertinent paperwork for the refusal. I will say that he was polite and extremely professional at all times. Although; I got the message loud and clear; we are now being tracked. Clearly; the company has embarked upon a 'campaign of intimidation.' If this campaign escalates further; I fear we will have some of the more junior Captains being bullied into accepting aircraft they might have otherwise not taken. I would respectfully ask the FAA representative of this committee to intercede on behalf of all airline pilots; and be proactive in addressing this blatant attack on Captain's Authority. That authority is the last and most important safety check that we pilots have to provide a safe product for our passengers and crew. To sit back and allow the erosion of this authority; in my opinion; could have disastrous results.