The atmosphere approximates an ideal gas, and as such you can relate pressure and density through the ideal gas equation. The form we use in meteorology uses mass density and is given by:
p=ρRTp=ρRT
where PP is pressure in units of Pa, ρρ is density in units of kg m-3, RR is the gas constant for dry air (287 J kg-1 K-1) and TT is temperature in Kelvin. This assumes a dry atmosphere and humidity will decrease density for a given pressure. Consideration for water vapor is usually brought in by changing temperature into virtual temperature TVTV where TV=T(1+0.61q)TV=T(1+0.61q) and qq is the mixing ratio of water vapor (units kg kg−1kg kg⁻¹).
Pressure increases with temperature because the particles have more kinetic energy (which is proportional to TT). Imagine a box full of bouncing balls, if these balls start moving faster the balls will hit the walls of the box harder, imparting more force on the box. Pressure is merely force per area, so if the force increases but the box stays the same size, the pressure has increased.
Air density can decrease with temperature if pressure is also decreasing. If pressure is constant, this cannot happen (they would be inversely related). Any time you specify a relation between any two of pressure, density or temperature you must hold the third constant or specify its behavior.
For example, hot air rises, but why then is it cold on top of a mountain. The answer is that hot air is less dense than the cold air surrounding it for a constant pressure, and being less dense it rises. With a mountain, the pressure is decreasing, and we likewise find in the atmosphere that temperature decreases with decreasing pressure.
