I've just had a thought.

You could do a similar thing with a normal engine. Take the hot exhaust gas, allow it to cool to room temperature inside a cylinder, and extract power from that piston as the cooling gas contracts.

That seems to be equivalent to what you're doing. It's called a 'bottoming cycle', and can get you a bit more efficiency, at the expense of extra parts. Many car engines actually do that, it costs more money and gives you a little bit more efficiency - called a turbocharger.

However all the bottoming cycles added up still can't give more than Carnot efficiency, which is still much less than 100% for burning gas and room temperature ambient.