The two cup trick is also used in Japan when making green tea, because the fragile tea leaves can be damaged if the water is too hot.
I once heard a tea master say that every time you pour the water from one cup to another, it cools down by 10 degrees (celsius). I'm not sure I fully believe him because pouring the water between cups 10 times didn't freeze the water. ;-)
It won't be a fixed "10 degrees". More like dt = (T_cup - T_room) x cooling factor.
I'll leave it as an exercise, as to what 10 pours will do, but it won't freeze the water; as I don't remember how to solve it and I don't have a piece of paper handy.
And that's not accounting for the specific heat in the state change.
Entropy is logarithmic with respect to degrees of freedom, and since degrees of freedom is (usually but not always) proportional to the energy of the system, the temperatures of the tea and of the room will approach each other faster when the difference between them is greater, but the heat exchange will slow down when the temperatures are similar.
EDIT I meant to say that entropy is proportional to degrees of freedom, which is logarithmically related to heat energy (except in some weird cases). Anyway, the conclusion was right :-)
I once heard a tea master say that every time you pour the water from one cup to another, it cools down by 10 degrees (celsius). I'm not sure I fully believe him because pouring the water between cups 10 times didn't freeze the water. ;-)