Bernoulli,
This is a case where you're trying to think too much. You're trying to outreason the altimeter.
Lets say your indicated altitude was 5,000 feet... when you enter the cold air the altimeter will indicate something higher like 5,300 feet. The altimeter does this because it thinks the air pressure is less because that is what an altimeter reats to... air pressure. Well, the higher the altitude gets, the thinner the air becomes...and the colder the air becomes.
If flying at 5,000 feet indicated, your altitude will not increase when entering a colder airmass. Your indicated altitude remains the same, and true altitude decreases, not increases.
Your indicated altitude remains the same when flying from hot to cold, but your true altitude decreases. One way of picturing this is to imagine a balloon. Stick a little model airplane on top of the balloon, and fix it in place with a little rubber cement.
Measure the distance between the airplane and the tabletop. This is your true altitude. Place the balloon in the freezer. It cools, it shrinks. Plut it back on the table. Now the distance between the little airplane and the tabletop is less...true altitude has decreased.
Same for a pressure drop. Instead of sticking the balloon in the freezer, just suck out a little air, and keep it the same temperature. Lesser pressure, true altitude decreases (assuming it's set to the same altimeter setting. In other words, if you change the altimeter setting to match the current pressure, your altimeter will read correctly. However, if you don't change it to match the current conditions, it won't read correctly. If it's set for a higher pressure, it's going to indicate higher.
You can also see this while sitting on the ramp. Simply dial in a higher pressure on your altimeter, and watch what it does. It shows an increase in altitude, even though you know you haven't moved at all. You can envision a pressure decrease as though the air mass you're in has "decreased" in volume...it's shrinking. If you're at a certain level or layer in that airmass, (sometimes we call this an altitude), the height of that layer is going down along with the rest of the airmass when the airmass "shrinks."
Your altimeter is based on pressure, and based on a reference pressure. If you're following a reference pressure level (altitude), and that altitude goes down, you go down...even though the altimeter says otherwise.
Don't confuse temperature lapse with altitude, with flying from one airmass to another. The altimeter is referenced to a pressure; it's not really that pressure we're flying in. It has no clue about temperature. For example, at 18,000', we're going to put 29.92 in the altimeter. It's not really 29.92" Hg up there, but we use that as a reference...that's the reference supplied to the little aneroid wafer in the altimeter, by compressing it using a spring. This is something you don't need to worry about; all you do is set the altimeter. Once it's set, the outside air temp is irrelevant if you have the correct altimeter setting in place.
When flying from one location to another above FL 180, for example, all aircraft are using the same altimeter setting. In any one given area, aircraft will actually be higher than 18,000 MSL, or lower than that, depending on actual temperature and pressure. However, everyone in a given area will be affected the same way...the important thing.
Where it does become an issue is at lower levels.
Cold air is not synonymous with high or low altitudes. Temperature generally decreases with an increase in altitude, but that has nothing to do with your question. With a decrease in temperature while flying at a constant indicated altitude, true altitude will decrease. Your question asks about getting colder with altitude, but your question must also presuppose flying at the same indicated altitude...vertical temperature lapse rate then has no part in the matter.
Your powerplant will perform better at colder temperatures, due to increased air density. However, with an increase in altitude, lower air density works against this performance increase. In other words, air pressure is less, though temperature decreases. Where a decrease in air temperature will provide increased engine and aircraft performance is an decrease in density altitude...not merely temperature.
