I think that the admin should be able to login into the live site as well when it's in maintenance mode. For example, if the admin is adding a new page to the site or changing a site layout they will need to preview what the change will look like on the live site, so instead of doing that while members are on the site, they know that they can work freely and if they have any errors at least it won't be seen by members until they take the site out of maintenance mode.
Hello dexignz
Good suggestion. In this way, when maintenance page is enabled, admin will be able to login only to admin area. Is it Ok?