> Mostly wrong.

Not treading ligthly I see. But you're wrong

> I think it is the first law of thermodynamics.

The first law of thermodynamics is conservation of energy in a closed system.

My thought experiment does not violate it. The closed system is the room in which the hot coffee resides.

There is no difference in the amount of energy but in the process in which thermodynamic equilibrium is reached.

At some point the coffee and the room will reach an equilibrium temperature that is very close to room temperature (bc its heat capacity is much higher than that of the coffee)

The coffee is drunk before equilibrium is reached (who wants cold coffee) so this is not about the energy content of a closed system, but about at which point of the state transition it is drunk.

First lets make a slightly different but simpler argument that proves that you are wrong: Lets assume the milk comes from a fridge, it therefore is colder than room temperature, while the coffee is hotter.

It should be obvious that allowing the hot coffee to give its heat to the room, then pouring the milk will be colder than pouring the colder milk into the still hot coffee resulting in the coffee heating the milk instead of the room, right?

I chose this example with cold milk because it makes it easier to comprehend that the milk has a heat capacity that it adds to the mix. With room tempered milk that is still true, its just not as ELI5.

Therefore both methods will eventually reach equilibrium temperature, but pouring the milk earlier will leave you with a warmer beverage at consumption