High oil temperature on an MD80 could not be controlled. Limits were exceeded during diversion to a nearby airport requiring an engine shutdown.

Narrative

About thirty five miles southwest of the nearest airport; I notice a left engine temperature high indication. At first; I reduced the power on the left engine while taking note of other engine indications. Oil pressure and quantity were normal. While at FL340 I told First Officer to request a lower altitude due to the reduction in airspeed. FL280 was assigned by Center. We ran the QRH for Engine High Oil Temp indication. The temperature varied from approximately 120 degrees C to approximately 154 degrees C. We tried to keep the temperature below the 135 degrees C/15 minutes by changing power settings to prevent an engine shutdown per checklist. We contacted Maintenance Control and they agreed on a diversion to the nearest airport. We coordinated the diversion with Dispatch and requested a diversion to the nearest airport with ATC. While entering the nearest airport airspace the temperature went uncontrollable above the 165 degree limit and we shut down the left engine per QRH. We landed the aircraft safely and taxied in to the gate using a single engine