A320 DIVERTS DUE TO A BURNING SMELL IN THE CABIN.

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THE CHIEF PURSER CALLED THE COCKPIT TO RPT AN ODOR WHICH SMELLED LIKE BURNT RUBBER. I ASKED HER IF THERE WAS ANY SMOKE AND SHE SAID NO. SHE HAD ASKED EARLIER DURING THE FLT A COUPLE OF TIMES TO WARM UP THE CABIN. IT WAS INDICATING A NORMAL 76 DEGS; HOWEVER; WE WARMED IT UP TO EVENTUALLY 80 DEGS. I SUSPECTED SHE MIGHT BE SMELLING HEAT FROM THE PACKS SO WE LOWERED THE TEMP. I HAD HER CALL ME BACK IN A COUPLE OF MINS AND SHE STATED THE SMELL HAD SUBSIDED. A FEW MINS LATER SHE CALLED BACK TO SAY THE SMELL WAS THERE AGAIN. I ALSO REAFFIRMED THAT THERE WAS NO SMOKE. AGAIN WE ADJUSTED THE CABIN TEMP. EVERYTHING APPEARED NORMAL WITH THE SYS. A THIRD VERY CONCERNED CALL CAME FROM HER. SHE STATED THAT THE SMELL WAS A LOT WORSE AND THAT ALL 4 FLT ATTENDANTS AGREED AS WELL AS A NUMBER OF PAX. THE SMELL WAS IN THE BACK AROUND ROW 22. EVEN THOUGH THERE WAS NO SMOKE; WE DECIDED THE WISEST ALTERNATIVE WOULD BE TO GET THE AIRPLANE ON THE GND. ZZZ WAS ABOUT 100 MI OFF THE NOSE AND WE WERE AT FL350. SINCE I COULDN'T BE SURE THAT A BURNT SMELL WOULDN'T ESCALATE INTO SMOKE; I DECIDED TO DECLARE AN EMER; HAVE EQUIP STANDING BY; AND PLAN A PRECAUTIONARY LNDG. I IMMEDIATELY NOTIFIED DISPATCH. WE ALSO REVIEWED ANY APPLICABLE CHKLISTS. WE DID A CABIN ADVISORY AND LANDED VISUALLY. I HAD THE FIRE DEPT INSPECT THE OUTSIDE OF THE ACFT AND LATER THE INSIDE AT THE GATE. MAINT REPLACED A HOT AIR VALVE AND WE DISPATCHED TO ZZZ2. I FEEL THE CREW USED GOOD CRM COM SKILLS. WE WERE WELL INFORMED AND FEEL THAT THE FLT ATTENDANTS ARE A VALUABLE TOOL IN RECOGNIZING ABNORMALITIES. WE ARRIVED 3 HRS LATE INTO ZZZ2; HOWEVER; RECEIVED NUMEROUS COMPLIMENTS FROM THE PAX.