It's a partial yes. You have to worry about pressure as well. You can burn fuel at a higher temperature by igniting the gas at a higher pressure. But there are certain constraints due to the fuel. If the pressure is too high, the fuel will spontaneous combust. That's why there are different octane rating of fuels and why higher octane fuel gives you better fuel efficiency (given your engine has some sort of valve timing like VTEC). This is also why Diesel engines tend to have better mileage than gasoline engines. Even though gasoline engines will perform better at the same pressure as Diesel, there just no way to bring the gasoline to the same pressure that the Diesel can handle.

When and how you combust the fuel can give you different fuel efficiencies as well. For instance, the Brayton cycle can achieve easily over 50% fuel efficiency. But in order to achieve this, you need a special housing (jet engines uses this cycle) to be able to compress the fuel to the specific pressure and also handle the pressure. Also, special fuel is required. These two things makes it not really practical for the standard consumer.

On a tangent, Internal combustion engines is probably one of the funnest and hardest class I took in graduate school...

You can, but it's worth pointing out that cars don't really run on the Carnot cycle because, unlike your air conditioner, they're not reversible. Usually they use something like the Otto cycle. To improve efficiency you can change your cycle or push the limits (temperature, compression ratio, etc.). The current generation Prius apparently uses an Atkinson cycle and achieves 40% efficiency. Some F1 engines notably have a thermal efficiency of 47%.

https://arstechnica.com/cars/2016/05/turbulent-times-for-for...

It's mostly about the max temperature. If combustion in a steel engine runs at 250 C and operates in a 50 C environment, the max efficiency is 1-325K/525K = 38%.

Ceramic, for example, can operate at far higher temperatures, so in theory a ceramic engine can be more efficient.

However, ceramic is harder to work with, and not as durable as steel, so it's not a practical alternative.

There's a good explanation and derivation of Carnot's results on the efficiency of heat engines in the Feynman lectures [1].

[1] http://www.feynmanlectures.caltech.edu/I_44.html

Yes; I believe this is why jet turbine engines have blades made of single crystals of exotic materials.