Altitude vs Temperature

People who ski, hike, run, or do other vigorous exercise at a higher altitude than usual typically feel some effect of the elevation.  It is common to feel difficulty getting enough breath.  Conversely, sometimes people who travel down from a higher altitude feel like they can run faster and longer than usual.  The cause, of course is that the lower density air at higher altitude provides less oxygen.  At a given altitude, though, the density of air also varies with the temperature.  In this post, we’ll look at the size of that effect.
In an earlier post, we looked at the standard atmosphere which gives average conditions for temperature, pressure and density as a function of elevation.  This figure shows the density of air as a function of elevation for the standard atmosphere.  

You can see that at 10,000 ft, the density is only about 75% of the density at sea level which is why skiers and hikers might feel out of breath.
At a given elevation, we could look at the effect on air density of perturbing the temperature by, say, plus or minus 20 °C.  This figure shows what that would look like along with the density of the standard atmosphere. 

It is easy to imagine that the change in density of air corresponding to a change in temperature at a given elevation could be correlated to a change in density associated with a change in elevation. 

That number turns out to be 35.9 meters/°C or 19.7 feet/°F (along the standard atmosphere line).  

This means that if you were out running your usual route on a day that was, say, 30 °F cooler than your usual running temperature, you’d get a boost as though you had gone down about 600 ft in elevation.  So, not a huge effect, but something to keep in mind when you are comparing summer and winter times.  Of course, other physiological effects of temperature are probably more significant…