Hot Start is, by definition, a start in which, max eng turbine temperature (EGT for a Jet or TGT for a turboprop) surpasses a certain limit established by a manufacturer for that type of engine for Start process (there are other Egt limits, for instance Max EGT for TO)..
It happen for several reason, nowdays, with start process being fully automatically controlled by FADECs, the must probable reason is a deteriorated turbine assy hardware (blades) or combustor chambers. Back then when start process was manually, as previously indicated, if fuel was added before or after the correct "fuel-on rpm" you may have a hot start.
Manufacturers give directions in accordance with EGT overlimit severity, example: Max EGT during start is "only" 5° C above the maximun acceptable level, pilot is allowed to continue a flight but should made an entry in the log book. Then maintenance will inspect the engine @ next station including a borescope inspection. If start EGT it's 15°C above limit, pilot should shutdown de engine and maintenance action is requiered before next flight.
The events in which there is an EGT, RPM, Vibration overlimits are automatically recorded and sent to maintenance by ACARS.