What was found
CLIMBING OUT OF KHPN ABOUT TO LEVEL AT 5000 FT WE RECIEVED A RIGHT RED ENGINE FIRE WARNING. BEFORE I COULD RUN THE CHECKLIST THE LIGHT AND WARNING EXTINGUISHED. I BELIEVE IT WAS FROM THE THEOTTLE RETARDING AT LEVEL OFF. NOTICED THE ITT ON THE RIGHT ENGINE WOULD FLUTUATE WITH THROTTLE POSITION, IT DID NOT MATCH THE LEFT SIDE. I SAW THE ITT IN THE AMBER, IM NOT SURE IT MADE IT INTO THE RED. WE RETURNED TO HPN UNEVENTFULLY. UPON INSPECT ON GROUND A BIG HOLE ON TOP OF COWLING BY THE OIL DOOR. REMOVED THE UNSERVICEABLE RT ENG AND INSTALLED A REPAIRED SERVICEABLE RT ENG. PERFORMED ALL REQUIRED OPS CHECKS WITH ENGINE WITHIN EMM PARAMETERS, NO LEAKS NOTED. MAINTENANCE ACCOMPLISHED IAW AMM 71-00-05 AND EMM 71-00-00.