2004 GS500F

Starts just fine when cold.
Once started, can run fine for a full hour, no noticeable issues.
However, once it gets warmed up and ran for a bit, if I shut it off, it has trouble starting. The warmer it gets, the more difficult to start, until the battery drains.
It will turn over a bit, but just not start, and eventually just clicks when I hit start.
If I let it cool off, it will start just fine.