What was found
RT ENGINE SHUTDOWN AT FL370 FOR AN HOUR AT CRUISE WITH NO ISSUE. STARTED TO CLIMB, AND ABOUT 2-3 MINUTES LATER AT ABOUT 39,500" HAD THE RT ENGINE OIL TEMPERATURE CLIMB HIGH, REDUCED POWER. IT KEPT CLIMBING AFTER STABILIZING FOR A SHORT BIT EACH TIME THE POWER WAS REDUCED. SHUT THE ENGINE DOWN, DESCENDED, THEN DIVERTED.