How are air temperature and atmospheric pressure interrelated?

Inherently, warm air is less dense than cold air, so it constantly tends upward. Thus, the higher it rises, the less pressure.

Cold air, on the contrary, is denser, due to which it displaces warm (lighter air) under its weight and tends to the ground.

According to this principle, vents are designed in houses. When warm air enters the room, it goes down, thereby raising the warm air up. Due to this, ventilation occurs.