While not as good at blocking off air as a door, a heavy floor length curtain does a pretty decent job. Look for "blackout" or insulating drapes. Assuming that you are dividing off your bedroom that has A/C from a house that does not have A/C (so you still get some insulation effect) rather than directly to outside like to a courtyard. It would be silent for the dog to go in and out just brushing past the drapes.
If your room has A/C and the other parts of the house do not have A/C, then my guess would be that the dog is trying to find a comfortable temperature balance. Gets too cool in the A/C so scratches to go into the living room, gets too warm in living room without A/C so scratches on the door to come back into the cool.
Dog might do fine in your room with a few blankets to tuck into or toss off, or depending on your comfort level have it very slightly warmed in your room than it is now. If you prefer the dog to sleep outside your room, consider a cooling mat or an elevated bed that lets air flow around the dog (Kuranda is a brand name of an elevated dog cot so you can see what I mean but there are several commercial options along with build-your-own ideas online)
