Support

There is no such reason in Admin Tools. Could it be "Admin Query String" instead?

If that's the case, your login session had expired before you clicked the Log Out button. This redirected you back to the /administrator/index.php URL without the secret query parameter which is what triggered the security exception.

It's definitely an expired or broken session. I know how this feature works and I can tell you so you have full knowledge of what's going on.

When you access the admin login URL with the special query string parameter (e.g. http://www.example.com/administrator/index.php?mysecret) Admin Tools sets a "flag" in the session. That's a fancy way of saying that it tells Joomla to store a variable in the session with the value 1. On every subsequent access to a URL under /administrator Admin Tools checks the session for the existence of this variable. If it exists the page load proceeds. If it doesn't exist or has a value other than 1 it throws the security exception you saw.

Obviously that would cause a problem on logout since the session is reset. Admin Tools handles a Joomla event at this point which allows it to redirect your browser back to the the admin login URL with the special query string parameter (e.g. http://www.example.com/administrator/index.php?mysecret) instead of the regular admin login page (e.g. http://www.example.com/administrator/index.php).

However, if the session has expired, reset or is otherwise broken when the logout event happens this redirection will NOT take place. Instead, the logout URL itself fails the check I mentioned above ("On every subsequent access to a URL...") and triggers a security exception.

Also note that if you are using the unified sessions feature of Joomla, where you have a single login session for the public frontend and the administrator backend, session expiration and breakage is more likely to happen. Moreover, using that feature exposes you to more risk regarding login brute forcing and login cookie stealing. While convenient it's bad for security and I recommend disabling it if you have it enabled.