The answer for internal combustion engines is a little more nuanced. The simplest answer is that for a race car engine, where you want the maximum power output with the throttle wide open, you get that with cold air because the cold air has higher density (at the same pressure), and so you can get more air into the cylinder and can burn more fuel. And the efficiency goes up because the fixed losses are a smaller percentage of the overall power.

For regular internal combustion used for transportation, you rarely have the throttle all the way open and are often running with the throttle way back from wide open, creating substantial throttling losses. Lots of work on improved engine efficiency is finding ways to mitigate that problem. Variable valve timing is one example. But another is actually to deliberately heat the intake air. The lower air density will reduce the amount of mass flow of air into the cylinder, enabling scaling back the power towards the lower level that's actually needed with the throttle open more, or ideally even wide open.

So warm air can result in better efficiency given a specific engine size and a power output requirement well below its capability, because it effectively degrades the engine power without degrading its efficiency. However, if you are just operating the engine at its maximum efficiency point, with a wide open throttle, perhaps because you are using it for something other than driving a car on public roads, you'll get better efficiency with cold air.

A little more detail and links to papers with a lot more detail are on this Wikipedia page:

https://en.wikipedia.org/wiki/Warm_air_intake