Yeah, there are actually doors in that archway. You don't remember them because they're usually all the way open, and thus inconspicuous. (There's even a sliding door in the window above the kitchen sink, so that the living room can be completely sealed off.)

Really, the problem there is the way the AC is zoned. There are only two zones, upstairs and downstairs. As I sort of vaguely alluded to in one of my comments, I would love to rezone so that the living room was its own zone, but that's not currently the case. So in practice, thermal isolation would only make my workspace *warmer* as things currently stand, since it is the most heat-absorbent part of the house...