A B737-300 IN CRUISE DIVERTED DUE TO A FALSE APU FIRE WARNING WHICH HAD BEEN DEFERRED TO ALLOW GND APU OP.

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ON JUL/XX/97 AT APPROX XA45 I RECEIVED WORD THAT ACFT XXXX WAS INBOUND FROM XYZ WITH AN APU FIRE DETECTOR INOP AND WOULD ARRIVE AT APPROX XB55. THE ITEM WAS LISTED AS A MUST FIX DUE TO ROUTING. THE ACFT WAS SCHEDULED FOR A 20 MIN OR LESS TURN. THE ITEM HAD BEEN DEFERRED BY BBBMM OUT OF XYZ. I INFORMED MY LEAD THAT IT WOULD NOT BE POSSIBLE TO FIX THE PROB IN THE TURN TIME AND HE SHOULD LOOK FOR A PLANE CHANGE. THIS WAS ACCOMPLISHED AND THE ACFT WAS REROUTED SO THE ITEM COULD BE DEFERRED. WHEN THE ACFT ARRIVED AT ZZZ I DID A QUICK OP OF THE SYS AND FOUND THAT IT DID WORK WITH THE EXCEPTIONS I NOTED IN THE TEXT OF THE LOG WRITE-UP. I FELT THE PROB; ALTHOUGH UNFAMILIAR TO ME; FELL WITHIN THE LIMITS OF THE MEL. I DISCUSSED IT WITH A LINE MAINT CTLR WHO FELT THE SAME WAY I DID. THAT IS THE ITEM COULD BE DEFERRED WITH THE APU LEFT IN OPERATION. THE CTLR CHANGED THE MEL REF AND I RELEASED THE ACFT FOR SVC. THE ACFT FLEW TO YYY. ON THE WAY BACK AT APPROX XZ30 INFLT WITH THE APU SHUT DOWN THE CREW GOT AN APU FIRE WARNING. THEY FOLLOWED PROCS AND DIVERTED TO ABC. AFTER AN INSPECTION AND FINDING IT WAS A FALSE ALARM THE ACFT CONTINUED IN SVC. BEING UNDER PRESSURE DUE TO THE TIME CONSTRAINTS; OUTBOUND CREW TO HAVE THE APU OPERABLE IF AT ALL POSSIBLE; CONSIDERATION FOR PAX OVERALL COMFORT DUE TO HOT WX IN YYY AND JUST WANTING TO ACCOMMODATE EVERYONE AFTER COMING BACK AFTER MY DAYS OFF; I OPTED TO LEAVE THE APU IN SVC. MY 34+ YRS OF EXPERIENCE SAYS THAT IF I HAD ANY QUESTIONS I SHOULD HAVE OPTED FOR COMPLETELY INOP'ING THE SYS. THE FACT THAT I WROTE THE EXCEPTIONS TO THE OPS TEST IN THE TEXT OF THE WRITE-UP INDICATES TO ME; IN HINDSIGHT; THAT I DID HAVE QUESTIONS. IN HINDSIGHT I LET MY USUAL HARD NOSE APCH TO THE OP SLIP AND OUTSIDE PRESSURES INFLUENCE MY JUDGEMENT.