Is the pop up message "Your session is expired"? I can re-create that condition if I access a Webmail link from a Cox webpage while I'm already sgned in to Webmail. Try closing all browser tabs and open Webmail with this link:
I can confirm I've never been timed-out after logging directly into webmail. I could log-on in the AM, not use it all day and it'll still be active when I shutdown my computer in the PM. The Cox Account page, however, will timeout on me.
Although I haven't been looking, I've never seen a webmail link on my Cox Account page. I think the closet I've noticed was email support or something similar.