B767-300 Captain reported uncontrollable right engine oil temperature at cruise altitude. The Flight Crew complied with the QRH procedures and performed an in flight shut down of the right engine. The flight crew requested priority handling and diverted to make a precautionary landing.

Narrative

A few minutes after we leveled off at FL370; it came to my attention our right engine oil pressure had fallen to the top of the amber band about 23 PSI; with the oil quantity at 0. Initially the oil temperatures were similar. We started by looking at the Engine Oil Pressure Checklist which indicated run until pressure is red. Then the oil temperature started to climb from around 110 up to over 140 so we started to look at the Engine Oil Temp checklist and coordinated with ATC and Dispatch for an upcoming divert to ZZZ. As the oil temperature reached 149 we ran the Engine Oil Temp checklist which has us bring the thrust back until the temperature goes down. We brought the engine to idle which initially brought the temperature down but about a minute later it started climbing again. Since the checklist directs a shutdown if the temperature reached the red line we elected to go ahead and shut the engine down and prepare for a single engine landing as we descended toward ZZZ; rather than be in a low altitude rushed situation for the checklist. We requested priority handling once the engine was shut down and asked for vectors. We advised Dispatch of the shutdown and priority handling request. We then completed all remaining checklist items and landed uneventfully at ZZZ on Runway XX. We had the fire support that was on standby check to make sure we were not leaking anything on the right engine then taxied to the gate. Using good CRM we were able to make the decision to proceed with a well planned shutdown that lead to an uneventful landing rather than a potential higher stressed engine shutdown.