There are two independent issues at play here - "Bowed Rotor Start" or BRS, and residual EGT.
BRS is exactly what it sounds like - differential cooling of the HP shaft causes it to bow. There is a window of between ~30 minutes and a few (up to six) hours after the previous shutdown where BRS can be a serious issue. Best case it will result in high vibes during start (which has been known to do long-term damage to engine components), worse case it will rub the compressor resulting in permanent performance loss, worst case it's been known to damage compressor blades.
A period of motoring before introducing fuel helps mitigate (but not totally prevent) BRS. Most engines have some level of BRS exposure, although the newer generation 2 spool engines are noticeably worse.
High residual EGT can cause hot starts - basically the residual heat in the engine heats the incoming compressor air and effectively causes the FADEC to over-fuel during start causing a hot start. So autostart is programed to motor the engine until the residual EGT is below some (engine dependent) EGT limit before introducing fuel.
While I don't have any first hand knowledge of the V2500, BRS is not an issue for in-flight starting since the HP rotor won't stop turning (if it does you probably don't want to restart it) so it doesn't bow.
The FAA has mandated something called "Quick Windmill Relight" (QWR)capability for the newer engine installations (for Boeing, it started with the 777). QWR is for a pilot that inadvertently shuts down an engine in flight and, realizing the error, quickly turns fuel back on. The FAA has mandated it needs to start and be capable of significant thrust within (IIRC) 90 seconds, effectively prohibiting residual EGT restrictions for in-flight starting.