Keep password protected page cookie even if login or logout from the site

Is there a way to keep the password protected page cookie even if user logout or login from the site.
I want to password protect some posts and i need the password timeout to be longer (like 1 month or event 1 year).
I extended this timeout with a snippet, but the problem is that when a user login or logout from the site this cookie(password protect) is no more saved and you need to write
it again.
The Idea is that i don’t want my clients to search always that password again and again but to be saved longer even if they login or logout as a user from the site.