I know things... A lot of things.
Join Date: Feb 2015
Motorcycle: GSXR 750 K4
Unless the battery is installed in such a way/location, that would significantly heat it up along with the bike, I can't see why it would be good when the bike is cold and bad when warm.
I'd be more inclined to ascribe the lower voltage while cranking when hot, to a higher resistance to the starter motor. (Since the motor sounds like it's turning slower, it's probably drawing less current, but multiplied with the larger resistance the voltage drop might still be higher).
Try inspecting the wiring between battery, starter relay and starter motor for loose or dirty connections and the like. Make sure to inspect the ground path too, if applicable.