gksudo doesn't support persistent sessions - normally the sudo command retains the privilege granted as a session which lasts 5 minutes or so. gksudo doesn't feature this, and requires the user to enter a password each time. While using gnome 3 (via ubuntu gnome), I have seen some programs which prompt the need for a password in a more modern, gnome-3 style, while having the behavior I described in sudo above. What is this program, or a program which has the features I require? Specifically - a GUI for sudo which retains the privileged session for some timeout.
There is no replacement; Graphical applications are not designed to and should not be ran as a different user. If an application requires root privileges it should be designed as multiple processes where a small minimal process can be elevated.
You would use Polkit at that point to launch the new process as root though.