Denser air calls for a richer carburetor setting regardless of how long the engine's been running. How much fuel is metered through the carburetor jet depends on the volume of air flowing through the venturi. The colder air is, the more oxygen molecules (and nitrogen molecules) there are in a given volume. More oxygen mixing with the same amount of fuel means a leaner mixture. So the carburetor needs to be set richer to get the same fuel:air ratio. And since it's the temperature in the venturi that matters, warming up the engine doesn't change things (unless you use carb heat, of course).
A cold engine has a different effect. Fuel vaporizes better when it's hot. When the engine's cold, some fuel pools in the intake, and some settles on the cylinder walls. Since less of the fuel that comes out of the jet actually burns, more is needed to get the same effective mixture. That's why carburetors have chokes.
Of course, cold weather means that the engine is colder when it starts (and perhaps really cold weather means that some fuel doesn't vaporize even after the engine warms up).