B767 CREW HAS DIFFICULTIES STARTING APU AT ALT AS REQUIRED BY MEL FOR ETOPS FLT.

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WHEN I RPTED TO THE ACFT FOR THIS FLT; MY FLT CREW AND I REVIEWED/DISCUSSED THE PLACARD ITEM IN QUESTION THAT REQUIRED THE APU TO BE RUN DURING THE ETOPS PORTION OF THE FLT. WE WERE ALL IN AGREEMENT THAT WE WOULD BE IN COMPLIANCE WITH THE SPECIFICS OF THIS MEL REQUIREMENT. ALTHOUGH I AM NEVER HAPPY TO FLY ACROSS ANY OCEAN WITH LESS THAN FULL COMPLEMENT OF EQUIP; I WAS SATISFIED THAT WE COULD FLY A SAFE OP. THE APU WAS RUNNING AT THIS POINT AND WE HAD NO DOUBT OF ITS OPERATIONAL STATUS. AS WE APCHED THE ETOPS PORTION OF THE FLT; WE ATTEMPTED TO START THE APU. I BELIEVE WE WERE AT FL350 AT THE TIME; AND THE APU FAILED TO COME ON LINE. WE HAD NOT YET REACHED THE COAST OUT POINT; WERE STILL IN CONTACT WITH ZZZ RADIO ON VHF; BUT NO LONGER RADAR CONTACT. WE HAD RECEIVED OUR OCEANIC CLRNC AND ALT/TRACK ASSIGNMENT. I BEGAN TO REALIZE THE DIFFICULTY THAT WE WOULD HAVE IN REQUESTING A LOWER ALT; BUT I BELIEVED WE ASKED FOR AND RECEIVED A DSCNT TO A LOWER ALT (FL330) AS THE OPERATING MANUAL SUGGESTS. FURTHER ATTEMPTS TO START THE APU WERE ALSO NOT SUCCESSFUL AT THIS TIME. IT WAS AT THIS POINT THAT I BEGAN TO REALIZE THAT WE EITHER HAD TO GET THE APU STARTED OR WE WOULD HAVE TO DIVERT TO BE IN COMPLIANCE WITH THE OPERATIONAL REQUIREMENTS OF THE MEL. WE CONTACTED DISPATCH AND TOLD HIM OF OUR DILEMMA. HE PUT A PHONE PATCH THROUGH TO MAINT. WE AGAIN DISCUSSED OUR SITUATION AND ASKED FOR GUIDANCE IN AN ATTEMPT TO START AND RUN THE APU. THE ONLY REAL SOLUTION WAS TO DSND EVEN LOWER. WITH TIME BEING CRITICAL; WE ASKED ZZZ RADIO FOR A DSCNT CLRNC TO BELOW THE TRACKS (I BELIEVE FL250). SEVERAL MINS LATER; ZZZ REPLIED THAT THEY WERE UNABLE OUR REQUEST. IT WAS AT THIS TIME; THAT I DECIDED TO EXERCISE CAPT'S EMER AUTH AND ADVISE ZZZ RADIO OF THE SAME. WE ANNOUNCED OUR INTENTIONS AND INITIATED THE DSCNT OFF TRACK PER NAT PROCS. SEVERAL MORE UNSUCCESSFUL ATTEMPTS TO START THE APU WERE MADE AS WE CONTINUED OUR DSCNT. I WAS JUST ABOUT TO DIRECT THE FO WHO WAS THE PF; TO TURN TOWARDS OUR DIVERT FIELD WHEN THE APU STARTED AND REMAINED ON LINE. WE ADVISED DISPATCH THAT THE APU WAS NOW RUNNING AND WITH HIS CONCURRENCE; I DECIDED TO ATTEMPT A RETURN TO TRACK AND ALT. AGAIN; TO RESTATE; WE NOW HAD A GOOD SHIP! WE WERE ABLE TO RECEIVE CLRNC TO THE NEXT POINT IN THE TRACK AND ADVISED DISPATCH THAT WE WOULD THEN AGAIN REVIEW OUR FUEL STATUS AT THAT TIME. AT THE NEXT RPTING POINT; WE CHKED OUR FUEL AND CONFIRMED THE OPERATIONAL STATUS OF THE APU. EVERYTHING APPEARED TO BE A GO AND IN CONCURRENCE WITH MY FLT CREW AND DISPATCH WE DECIDED TO CONTINUE TO DEST. THE APU CONTINUED TO OPERATE AND FUNCTION NORMALLY. AT COAST IN POINT; WE TURNED THE APU OFF SINCE IT WAS NO LONGER REQUIRED PER MEL. AFTER WE LANDED AT ZZZ2 WE STARTED THE APU FOR ENG SHUTDOWN AND AGAIN IT STARTED AND FUNCTIONED NORMALLY. I VERBALLY APPRISED THE OUTBOUND CAPT/FLT CREW OF THE STATUS OF THE APU AND BRIEFLY DESCRIBED OUR ACTIONS ON OUR FLT. SINCE HIS ETOPS PORTION OF THE FLT WOULD BEGIN ALMOST IMMEDIATELY AFTER TKOF; HE WOULD BE ABLE TO ASSURE THE OPERATIONAL STATUS OF THE APU BY LEAVING IT RUNNING; AND THEREFORE ASSURE COMPLIANCE WITH THE SAME MEL REQUIREMENT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR STATES THAT THE FAA HAS DECIDED NOT TO PURSUE ENFORCEMENT ACTION AND THAT HE OPERATED IN A PRUDENT MANNER. HE HAS LEARNED FROM THIS EXPERIENCE AND WILL NOT BE CAUGHT OFF GUARD IN THE FUTURE.