I'm having an issue with Magento CE version 1.9.0.1 with admin session timeouts. It goes like this:

1. I log into the admin.
2. I go to the Catalog page.
3. I leave the window open, and after a while my session times out.
4. Later, I come back to the open window, and click on link to manage categories.
5. Because my session has timed out, I see the login screen. I put in my username and password and submit.
6. The page refreshes and requires me to login again. I put in my username and password and submit.
7. The second login works.

This is different from the infamous "can't log into the admin, page just refreshes" bug because it only requires two logins to work every time.

I think it might have something to do with an old key being left over in the URL on step 5, but I'm not sure. Any ideas?

**Edit 1**
To be clear, I am not looking to simply extend the lifetime of the session.

**Edit 2**
I should note that I have a different domain for the admin than I do the frontend. I can't find a way to set a different cookie domain for each.