Yes. Air is denser when it is colder. But it is denser on both sides of the sail, and it is the pressure difference that creates the force.
Imagine extremes to make the comparison between light and heavy air understandable. First think of a sail in space. Space is not 100% empty, it has just very, very little particles. Probably you are not gonna see anything hapening to your sail if these particles are moving at 12 knots? Now imagine using a sail in water with a 12 knot current. Water here represents very very heavy air. Probably your sail rips to pieces pretty soon.
So i dont know for certain how to explain scientifically why i think your airpressure difference thingy is wrong, but it really seems wrong to me