Not a chemist, so could not quote thermal expansion coefficients of mixed gases; I do know a nitrogen filled tyre, inflated in cool Northern hemisphere and pressure checked in a warmer climate, has increased by near 5%, conversely a drop the other way around; IMHO air would not be noticeable different. New tyres will initially loose pressure, they are more elastic expanding during running-in and loosing perhaps 5% pressure. As above, no real benefit for the Joe Blogs driver.
F1, think it is more a safety aspect for fire suppression.