cool air DOESN'T mean more oxygen.
actually, from what i recall, sucking in cool air will result in a greater amount of oxygen taken in during the intake stroke. because any matter that is cooled becomes denser, which is a result of the the molecules becoming more compact. the same principle explains why sealed cooking pans, rice cookers etc. all expel air at high speeds when heated, the heated matter expands due to the decrease in density increasing the amount of pressure in a given unit of volume, and if you cool the matter the opposite hapens. the matter becomes denser and therefore becomes more compact. More compact matter means that there will be a greater amount of matter per unit of volume than matter that is less dense.

Remember, as you climb/drive to higher altitudes you get less oxygen but air gets colder.
in truth, the amount of air in the atmosphere is directly proportional to altitude. so the higher you go the thinner the air becomes. it is true that the amount of oxygen is less when you are at higher altitudes but this is not because the temperature is lower. it is in fact due to the increasing altitude.
The troposphere is the lowest layer and contains about 75% of the total mass of the atmosphere, but the thermosphere, which is one of the two outer layers contains less than 1% of the total mass of the earth's atmoshpere. The thermosphere's temperature could reach 441 degrees farenheit, which is very hot. If you believe that the amount of air decreases when the temperature decreases then this layer should have the largest percentage of mass of the atmosphere. But it doesn't. Air is closer to the lowest layer because of gravity.

So colder oxygen means more oxygen per unit of volume.