There is a thermodynamic formula for gases that some of you might like to know. You can read about it in physics books or some places on the web. It is called the ideal gas law. Here is one place to learn more about thermodynamics:
Thermodynamics - Wikipedia, the free encyclopedia
Here is the ideal gas law:
PV=nRT
P: pressure
V: volume
n: number of molecules (related to the mass of a gas)
R: a gas constant
T: absolute temperature in degrees Kelvin
If the absolute temperature of an ideal gas doubles while the volume stays constant, the pressure will double. If the absolute temperature doubles while the pressure stays constant, the volume will double. In an engine, the pressure of the gas trapped in a cylinder is raised about three to four times when the gasoline burns. It might go from 180 PSI absolute pressure to about 630 PSI when the piston is near the cylinder head. I guess the absolute temperature goes up about three to four times also. There may be changes in the n factor (number of molecules) during combustion because of chemical reactions. The fuel mixture is not an ideal gas but most gases behave reasonably similar to an ideal gas.
It's not clear what intake air temperature is best. A higher temperature reduces pumping loss by requiring a larger throttle opening for the same power. It also increases the speed of combustion. On the other hand, a higher air temperature will increase heat lost to the engine. I suspect that the best temperature may be higher during light loads than at heavy loads. I know that old thermostatically controlled air cleaners would deliver cool air to the engine at full throttle but that may have been to help power instead of fuel economy. If the engine uses water injection, the best temperature might be higher than if it doesn't. I think that testing an engine's efficiency is the best way to determine what air temperature works best. It may not be the same for all engines.
The pressure increase is greater when the fuel mixture is colder, so power is increased. Fuel used is also greater because of increased density of cold air. There is a formula to calculate the theoretical efficiency of an Otto (gasoline) engine. It says that the efficiency can be calculated from the compression ratio. The air temperature is not in the formula. You can read about the formula here:
The Otto Cycle: Compression Ratio vs. Efficiency
It includes a chart. It shows the theoretical efficiency would be 50% with a compression ratio of about 5.7. It would reach 60% with a compression ratio of 10. The actual efficiency is less because of many factors that cause energy losses.