Lets examine some air pressure data from the Gulf of Mexico. The following data was measured at Station 42001 in the central Gulf. As you can see, air pressure does vary according to temperature.
Cold air is more dense than warm air, i.e., it weighs more.
As a result, it tends to sink. Warm air, on the other hand, is less dense. Therefore, it weighs less and tends to rise. Meteorologists say that warm air is buoyant. Look at the above image.