B757 DEPARTED WITH INOP OIL QUANTITY INDICATION. DURING CLIMB; HIGH OIL TEMP RESULTED IN ENG SHUTDOWN AND DIVERT.

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DEPARTING ZZZ1 HAD MEL ON L ENG OIL QUANTITY INDICATOR INOP. WE CHKED WITH ZZZ1 OPS THAT THE OIL QUANTITY HAD BEEN SVCED; THEY SAID IT HAD BEEN DONE ON THE OVERNIGHT. WE ACCOMPLISHED ALL CHKS; DEPARTED ZZZ1. WHILE CLBING THROUGH FL320 TO FL340; THE FO NOTICED THAT THE L ENG OIL TEMP WAS CLBING; AND THE INDICATION TURNED RED. I CALLED FOR THE CHKLIST AND REQUESTED FROM ATC A DSCNT TO FL300. THE CHKLIST LED US TO AN INFLT ENG SHUTDOWN. WE ACCOMPLISHED THE ENG SHUTDOWN; AND DECLARED AN EMER. I CALLED DISPATCH AND HE PUT ME IN TOUCH WITH MAINT. IT APPEARED THERE WASN'T MUCH MORE WE COULD DO AND I REQUESTED WX FOR ZZZ2 AND ZZZ3 FROM DISPATCH SINCE WE WERE ABOUT EQUAL DISTANCE BTWN THE TWO. DISPATCH SAID THAT ZZZ3 WX WAS OK AND HAD 10 MI VISIBILITY. I DECIDED TO GO TO ZZZ3. FROM ATC WE REQUESTED A TURN DIRECT TO ZZZ3 AND A DSCNT TO FL240. WE ACCOMPLISHED ALL CHKLISTS. I BRIEFED THE FLT ATTENDANTS AND TOLD THEM TO DO A CABIN ADVISORY. I BRIEFED THE PAX ON THE PA THAT IT WAS A PRECAUTIONARY ENG SHUTDOWN; AND WE WOULD BE DOING A NORMAL APCH AND LNDG AT ZZZ3. APCH AND LNDG WENT UNEVENTFULLY; WE TAXIED CLR OF RWY. WE WERE TOLD TO TAXI TO REMOTE PARKING. WE ACCOMPLISHED SHUTDOWN CHKLIST; AND HAD THE ACFT ON APU PWR. WE HAD EXCELLENT FIRE AND RESCUE SUPPORT. AFTER THE FIRE DEPT DEEMED THAT EVERYTHING ON THE OUTSIDE WAS SAFE; A SET OF STAIRS WERE BROUGHT TO THE ACFT AND WE DEPLANED THE PAX ONTO BUSES TO THE TERMINAL. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: REPORTER STATED THAT THE HE WAS STILL UNAWARE OF THE STATUS OF THE OIL QUANTITY AFTER THE EVENT. HE HAD RECEIVED NOTIFICATION THAT THE ACFT WAS AWAITING AN ENG CHANGE AFTER THE PRECAUTIONARY LNDG.