Ok, depends if we are talking TGT or EGT. Also upon the particular engine design as to where EGT sensors are. I accept that an EGT temp rise might be indicated after engine shutdown, but the thread started with a discussion of the starting case. If the engine has not lit properly and the unburned fuel is igniting in the exhaust duct, I would expect that any sensors would show a lower than normal temperature, since any combustion in the engine is inefficient at that time. The motoring engine would also be blowing the heat out of the engine. I have operated many different engine types where the 'hot start' - an internal starting fault - is readily seen on the gauges , whereas a jetpipe fire is not.