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. Imagine that the little circles in the beakers are air molecules. The image on the left shows the beaker of molecules when the air is cool. However, the image on the right shows the same beaker when it is warmed. As you can see, some of the molecules leave the beaker when heat is applied.