(it's not a coincidence, it's the exact opposite formula. You can only get 20% out of a certain heat engine because running it in reverse would be a 500% efficient heat pump, and the closed loop can't be more than 100%. This still applies even if the engine isn't built to be able to act as a heat pump, because some other engine could.)
I don't see why that would be the case. Why is the 20% of eg. an internal combustion engine coupled to the 500% COP of a heat pump? I see a vague connection as an ICE also uses expanding gasses etc, but that's about it. What am I missing?
An ICE is not an ideal heat engine. But if you did have an ideal heat engine, it could either produce work by reducing a temperature difference, or increase a temperature difference when provided with work. No physical device can do better than an ideal heat engine.
