FATIGUED AND HARRIED BY ON TIME PRESSURES; CAPT OF CARJ FAILS TO ENSURE PROPER MAINT IS PERFORMED ON AIR CONDITIONING SYSTEM ANOMALY WRITTEN UP ON FIRST LEG PRIOR TO SUBSEQUENT DEP.

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I NOTICED WHILE CLBING TO FL390 THAT THE L AIR CONDITIONING PACK WAS CYCLING ON AND OFF. ALL INDICATIONS ON THE R AIR CONDITIONING PACK WERE NORMAL UNTIL THE R SIDE PRSOV STARTED TO CYCLE CLOSED AND OPEN. AT THIS POINT WITH BOTH AIR CONDITIONING PACKS CYCLING; I INCREASED THE TEMP ON THE R PACK; THE INCREASE IN TEMP SOLVED THE R SIDE PRSOV FROM CYCLING; THE L SIDE AIR CONDITIONING PACK CONTINUED TO CYCLE ON AND OFF. I THEN INFORMED THE CTR CTLR THAT I WAS HAVING AN ISSUE WITH ONE OF MY AIR CONDITIONING PACKS AND REQUESTED FL310. WITH 1 AIR CONDITIONING PACK WORKING NORMALLY AND 1 AIR CONDITIONING PACK CYCLING; I MADE THE DECISION TO MAINTAIN FL310. I INFORMED OUR MAINT CTLR VIA ACARS THE STATUS OF THE ACFT INCLUDING A DESCRIPTION ON THE AIR CONDITIONING PACKS. I WROTE THE L AIR CONDITIONING PACK IN THE #1 DISCREPANCY COLUMN AND ONCE AGAIN INFORMED MAINT CTL VIA ACARS THAT I HAD WRITTEN UP THE L AIR CONDITIONING PACK IN THE ACFT LOG. I THEN SENT AN ACARS MESSAGE TO DISPATCH AND REQUESTED A NEW FUEL BURN FOR FL310. ALL INDICATIONS ON THE PRESSURIZATION SYS WERE NORMAL AND THERE WAS NO DANGER AT ANY TIME TO THE PAX AND CREW ON BOARD THE ACFT. UPON ARR AT THE GATE IN ZZZ; I WAS UNDER A TIME CONSTRAINT TO GET THE NEXT FLT OUT ON TIME. I ALSO PICKED UP A NEW FLT ATTENDANT; IN ADDITION TO WHICH I HAD TO FILL OUT A NEW PAGE IN THE ACFT LOG. WITH A TIME CONSTRAINT; A FLT ATTENDANT SWAP; A NEW LOG PAGE FILLED OUT; FATIGUE; DIFFICULTY RETRIEVING THE DISPATCH RELEASE; AND THE FACT THAT THE WRITE-UP WAS ON THE PREVIOUS PAGE AND NOT IN SIGHT; IT WAS LEFT AS AN OPEN MAINT DISCREPANCY.