I have the Star Wars Xbox 360, and just about a week ago it started turning itself on randomly throughout the day, followed by it also turning itself off then back on again a few times within a 3-5 min period. At that point I go and unplug the system.
Also, when I try and use the system, it will do the same thing 25% of the time, anytime from right when I power it on, to about an hour. The longest stand alone instance Ive been able to play is about 2 hours. After the first few occurrences, I moved a small fan near the Xbox to help keep it cool, even though the Xbox is not inside of anything, or within two feet of my other electronics. It sits on a wood coffee table.
There are no LED indicators, the power supply stays green when on and switches to orange when off, and I've done this with both using the the power button on the front of the Xbox as well as both a wired and wireless controllers to see if any of the effects changed anything. All of which had the same outcome.