Long cold starts are the worst, worst case scenario is a "hung" start, typically on PT6`s. It stops accelerating at say 35% and you can then watch the temp slowly climbing.
Another example of this effect is when you load at small turbine (King Air etc) with generator and aircon. The compressor rpm goes down and the temp goes up as the engine is running on metered fuel - solution on Kingair push up idle on others crack the thrust lever/throttle to increase the Compressor rpm and cool the motor down..