BUt this logging out wouldn't have to do with the cookies? Don't know much about this.
I think WoT has it. It sounds like your browser is emptying the cookies after you close the window. It needs to have those in order to recognise the website when you return.
Each browser is slightly different so you may have to poke around in the Tools menu to find which settings need tweaking.
For example, this is where to find it in Internet Explorer
Tools > Internet Options > Advanced
Security
Empty Temporary Internet Files folder when browser is closed - uncheck