On certain types of engines (the PT6A family comes to mind), the starting temperature limits are actually significantly higher than the takeoff or MCP limits.
As others have already mentioned in replies above, there are a great number of variables that will affect what the indicated temperature limits during different phases of engine operation are.
One, of course, is where exactly the temperature sensors are located. Typically, they won't be located at the hottest part of the engine, therefore there is a relationship between measured temperature and the critical temperature at another point. This relationship changes depending on the nature of the engine operation - start, idle, or high power.
Next is the lag inherent in the measuring system. The measuring system is usually optimized for high power settings, and therefore might not react quickly enough to catch peaks during starting - thus resulting in a reduction in the indicated temperature limit during a start cycle.
Last, on certain types of engines, much higher temperatures can be tolerated during a start because the engine parts are not yet heat-soaked, and the rotation speeds are low. By way of example, you can pass your hand through top of the flame from a candle and not burn yourself - providing you don't leave your hand there long enough to let it become heat-soaked.
I think it would be very difficult to provide an answer to your question that is valid for all engine types.