Sorry for my part in the confusion.
I was talking mostly about jet turbine with air powered starters. Starter torque will not change in dependance of batteries' power.
And concerning residual EGT values after a short turnarounds, again manufacturers establish the maximun residual EGT value before starting an engine and, if it's a modern eec engine, should computer (eec) senses a higher than normal EGT before start, it will only cranck the engine, without fuel injection and ingnition power thus "blowing" the engine and reducing the EGT for a while, then computer will stop the engine and, after engine is fully stop, a new start attempt, this one with fuel and ignition will follow.
In older engines w/o eec, there are strict residual EGT margin before attemping a new start. Should you follow them and there are not engine malfunctions, start process should proceed flawlessly.