Moto man posted this problem and I have the exact same problem hoping to find the fix.
(O.K. Here's my problem, 07 S4RS, 15,297 miles on it now, I can crank the bike, run it to temp (even 220 F) it will idle and run fine. I can even ride it, no problem. However, if I turn off the ignition (when hot, above 170 - 180 F), then crank it back up, it will not idle. Must give it throttle to keep it running. Let it cool to room temp, and it runs fine again.
I have checked: Battery, charging output, fuel pressure, spark plugs, coils, TPS setting, air & engine Temp sensors readings seem to be normal.
I'm stumped at the present time. Anybody seen this before/ got any ideas?)