I'm running 9.0.2. (stupidly upgraded to 9.0.1 thinking that might solve the issue)
(9.0.2?)
Try these steps in order:
1) Try a hard reset... hold the Home and power buttons down until the "power off" slider appears, then keep holding... the phone will summarily turn off. Wait a beat, then press the power button again. The phone will begin a lengthy reboot process. If you had a rogue process running (it happens), this might kill it. Then go on your way and see if the problem recurs.
2) If the problem persists, connect to iTunes and do a complete restore-from-backup. Same theory: a corrupt process may be running, and this should kill it and prevent recurrence.
3) If that still doesn't solve the problem, make an appointment at the Genius Bar and bring the phone in, making it clear to them that you've tried the two steps I just mentioned. They can run some additional diagnostics to see what's up. Could be a hardware issue.
Something similar happened to my 3G when it was still in warranty over a year ago. A fully-charged battery would run down in a few hours, and the phone was warm. I tried (1) on my own; the Genius had me go home and do (2), and when I came back in they just gave me a new phone.
Oh, there's one more thing. You may have a power-sucking application that's running because you told it to. If you're running iOS 4.x, do a rapid double-click of the Home button. Up will pop a slide-able list of all the apps you've used recently. Some of these may still be running, and if even one is a power hog (such as a navigation application) then that might explain your issue. Kill unwanted apps by tapping-and-holding their icon in the slide-able list until they wiggle and a little red X-in-a-circle shows. Tap the X-in-a-circle. App killed. ...These are early days in smartphone backgrounding and programmers are still learning how to deal with the power issues, so you might need to be a little bit vigilant about it.