You would need a tight sealed door to the structure for efficiency reasons. So because of that a barn door would be least efficient.

If the door doesn't seal tight when closed, you will lose heating and cooling thru the openings of the door.

If you want a sliding door you could use a patio style all glass door.

The more glass you have the least efficient it is, but if it is out of the way of the sun via sunlight infiltration it should be ok for most uses.

Could you provide a photo of the space?

It looks like the arch in the first photo, without any door. We'd probably build a new door frame to seal tightly with the door we choose.

You could install a swing patio door. Anderson has french style doors but they are rectangular in shape so you would have to give up the arch. Marvin Windows has french style glass doors with an arch. I am not sure if you can specify the shape of the arch. It is probably cheaper to redo the opening and make it a standard rectangular size.

I had this problem two years ago with door between a sun room and family room and kitchen. I ended up installing the Anderson 200 Narrowline doors in 2 and 4 panels. The patio door is designed for exterior installation so it very good for sealing and insulation. The drawback there is a threshold that you have to step over when going through the door. If you redoing the opening you could probably lower the door the minimize the height of the threshold above the finished floor.

I used an exterior door for this issue. Andersen swing wood door, full glass, vinyl-clad on the unheated staircase side (entry vestibule), stained on the living room side. Proper "compressible" whether stripping around jamb, never had any moisture on the glass.