Running Firefox as root in a regular user's session is not supported. ($XAUTHORITY is /run/user/1000/gdm/Xauthority which is owned by radix.)

earthmeLon's warning is very important, and you should try to solve your problem without having to run Firefox as root.

However, if for some reason you really need to do that (I won't judge you), you can do this:

sudo -i

To start a root shell, then make a backup of root's .Xauthority if needed

test -f .Xauthority && mv .Xauthority .Xauthority.bak

Copy the file from your home to root's

cp -a /home/youruser/.Xauthority .Xauthority
chown root: .Xauthority

Then you can start Firefox with:

XAUTHORITY=/root/.Xauthority sudo firefox

Remember to exit the root shell when you've finished with it.

Also remember to unroll your .Xauthority backup when done:

sudo -i
test -f .Xauthority.bak && mv .Xauthority.bak .Xauthority
exit