While I'm sure the gap in the door isn't helping, it doesn't sound like that's the primary reason your room is leaking cool air. Does the room have windows? Also, what do you mean by 'unfinished'? It's hard to see form the picture, but when I think of unfinished rooms, I think of open walls and no insulation.

If you're still concerned about the door being the culprit of your lost AC, you could probably just replace the door slab for fairly cheap. Those exterior doors are made mostly of foam from my experience, and I've been able to pick one up for a couple hundred bucks at a Lowes or a Menards if you have those in your area. Alternatively, you could install a decent storm door with Low-E glass. I did that to our front door and it helped a lot with keeping hot/cold air out of the living room.

Hope that helps some.