ACR MLG BRAKE INDICATED HOT DURING CLBOUT AND CAPT FAILED TO MAKE ACFT LOG ENTRY AFTER DSNDING AND EXTENDING THE GEAR TO COOL OFF THE BRAKE.

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THE INCIDENT OCCURRED ON CLBOUT FROM MSP. A MASTER CAUTION MESSAGE WAS RECEIVED INDICATING THAT #3 BRAKE TEMP WAS AT 680 DEGS. A BRIEF DISCUSSION WITH THE FO ENSUED AND WE BOTH FELT IT WAS AN ERRONEOUS MESSAGE. TO BE CONSERVATIVE WE OBTAINED A DSCNT CLRNC TO 10000 AND EXTENDED THE LNDG GEAR IN DSCNT. SHORTLY AFTER EXTENSION THE AMBER 680 INDICATION CHANGED TO GREEN AND 80 DEGS. WE CLBED BACK UP TO CRUISE ALT; CHKED FUEL; CONTACTED DISPATCH; AND PROCEEDED ROUTINELY TO SNA. LNDG AT SNA; THE #3 BRAKE TEMP INDICATOR OPERATED NORMALLY; PROVIDING TEMP INFO SIMILAR TO THE OTHER 3 BRAKE INDICATORS. IN OTHER WORDS; THE AIRPLANE WAS OPERATING NORMALLY; THERE WAS NO SQUAWK OR PROB TO WRITE UP. THERE HAD BEEN NO OTHER MESSAGES OR ANYTHING NON-ROUTINE DURING THE 3 HRS OF REMAINING CRUISE TO SNA. I MENTIONED THE INDICATOR PROB TO THE WAITING MECHS (DOING A DAILY CHK I PRESUME); BUT MADE NO LOGBOOK WRITEUP. I DID MAKE THE EFFORT TO CONTACT THE FLC TAKING THE AIRPLANE OUTBOUND TO NOTIFY THEM OF THE FALSE WARNING. IN RETROSPECT; I CAN SEE WHERE THE ARGUMENT COULD BE MADE FOR A LOGBOOK WRITEUP SO AN OFFICIAL RECORD COULD BE BUILT IN THE COMPANY COMPUTER SYS. THERE WAS NO HISTORY OF SIMILAR PROBS ON THE AIRPLANE. MY CONCERN IS THE LACK OF A LOGBOOK WRITEUP.