How does temperature change air pressure?
An increase in temperature causes an increase in air pressure, everything else being equal.
An increase in temperature increases the kinetic energy of the air molecules. An increase in kinetic energy cause an increase in the average speed and momentum of the air molecules. Molecules that are moving faster with more momentum collide with other molecules with more force creating more pressure.
There is a direct relationship between temperature in degrees Kelvin and pressure. So if the temperature increases the pressure increases (Gay Lussac's Law).
However, temperature is directly proportional to volume (Charles' Law) and volume is indirectly proportional to pressure (Boyle's Law). The end result of this is that if a parcel of air is heated and it can expand (volume increases) then the pressure will go down. So the air over a black asphalt parking lot (absorbs all the energy from the sun) beside a lake (reflects some energy from the sun) can easily expand into the area over the lake as that air is not being heated.
When we look at the equator, all the air is being heated and all of it wants to expand. This causes the air pressure over the equator to be higher than the pressure over the poles.