Really sorry to hear about this issue with the Xbox 360 App.
As there are so many reports of this issue it has been escalated to our development team.
Once I receive an update from the team I will update this post.
I'm going to move your post into the thread that has already been setup regarding this error.
There is an update in that thread regarding the issue that you are experiencing on your Xbox 360.
This is happening to us too. I have deleted the app off the xbox360 and downloaded it again several times but it keeps logging me out every time I select the app. I have paid for a service I can't use. Sometimes if you leave the Xbox run for 5-10 mins before you try it it works but then disconnects from Xbox live after 5mins of the programme. When the glitch is fixed, please let us know. Can I get a refund or free month please as I can't access the service I've paid for?