there is a site that i frequent and the password was changed on me but strangely i can still access the site...i am attributing that to the fact that i have the cookie store don my firefox browser (when i try to login with my safari browser with the same password it does not work anymore)

is there any way i can ensure that i do not get logged out?
how do i make the cookie permanent? i know that it sometimes "Resets" itself on its own how do i preven tthat?