Air does indeed flow from high pressure to low pressure area (see the wind arrows on a weather chart), but in the case of two rooms the much more important effect is that of warm thinner air rising towards the ceiling when the air from the two rooms gets mixed.
Thus, cold air from the cold room will be leaving the room close to the floor (if the temperature difference is large enough you can actually feel it, otherwise you can use a candle to detect the direction of air movement). At the same time, warm air from the warm room will rise and move into the cold room close to the ceiling (again, sometimes you can feel this otherwise you can detect it with a candle). Similar air movements take place between your house and the outside when you open your house's door in winter or summer.
The tendency of warm air to rise towards the ceiling is actually exploited in floor heating. This was understood and taken advantage of already by the ancient Romans, see hypocaust.