Well, then this is one of the rare ones. This room has NO door. There isn’t even a door frame (or anything like it). The room is designated as an office – hence they got away without putting in a door or door frame.
One of the things we used to do when we lived there was to place a box fan on the floor in the hall about 4 feet in front of the entrance to that room, blowing the cooler air INTO that room. This seemed to help, but didn’t completely solve the temperature differential.
J.
Ch-ching, the price of a fix is going up. Sorry, dude. Is there space above the doorway to add a fan to pull hot air out of the room? You can add a switch to reverse the air flow to see which works better? Either way, that will require running AC to the fans location. How handy are you?
If you decide to go with adding a cold-air return duct, be sure the register is as high on the wall as possible. That’s where the hot air is. You’ll also need to figure out how to add the new duct work to the old cold-air return duct. If the furnace is directly below the hot room, you may only have to tear up a wall in the hot room. If not… well, you’re going to need access to add and fasten the duct work. It is what it is.
As it stands now, the doorway acts as your cold-air return. As you’ve noticed, that’s not very efficient. It’s easier to cool a room with a closed door. If you chose to add a cold-air return duct or fan, you should also consider adding a door.
You should also consider the other suggestions offered up thread. Room-darkening shades, insulated shades, insulated window coverings, etc… Those will reduce direct, and indirect sunlight. Indirect sunlight being any sunlight reflected off adjacent buildings, or hills.
Personally, it sounds like a fun, but time consuming, project.
You’ve misunderstood the situation. 
No way in hell I’m going to do this work. I remember when I was a young whippersnapper and had to do all of my own work on my Volkswagen. I remember thinking right then that as soon as I wasn’t so poor, I was going to pay other people to do all the work.
That was decades ago. And I’m no longer poor. 
The furnace / AC is on the (flat) roof of the town house. The ducting and register in the room are installed in a sheet-rocked section hanging from the ceiling.
So the register is on a vertical section of sheet rock just below the ceiling. My concern about putting in an air return (in the ceiling) is that the register is going to blow the air across the ceiling and be immediately sucked up by the air return. Not efficient.
I’m going to have to wait for the HVAC guy to come out and analyze the situation. I wanted to get some comments about the potential problems and solutions. This has all been a big help!
J.