Ohhhh this is a pretty straight forward fix :)
Every time you go near the gate, work lots. Circles, changes of direction, leg yield, figure eights. When you go away from the gate, relax, loose rein and just hanging out.
Back to the gate, work hard, away from the gate relax.
Sky used to do this with me. It lasted about.. a week and then I fixed it.
I like Cowchick's approach too, work hard outside arena, relax inside.
Or you can do what we do at my barn. We leave the arena gate open and we change from outdoor to indoor, or if you don't have that set up, in the arena, out of the arena.
"Strength is the ability to use a muscle without tension"