Alexban,
I'm not sure you're reasoning is completely correct.
How 'hot' something is must be in reference to absolute zero. (-273 oC). In your distance example you are choosing an arbitrary origin, which is different to the Kelvin temperature scale, as this origin is defined and absolute.
The confusion arises by introducing an arbitrary origin, which is as you say, the triple point of water. However, it is perfectly possible to have chosen different reference points (e.g. Farenheit where 100F is body temperature, give or take a few degrees).
E.g. say it was 80 oF on a summers day (approximately 27 oC), if you were wondering what it would be like to be twice as hot would you say that it would be 160 oF (71 oC) or 54 oC (130 oF).
As you can see the choice of origin vastly affects the result.
Therefore, the only true sense of 'twice as hot' must be in reference to absolute zero. Twice as hot in this sense means that particles have twice as much kinetic energy as they did before. (As absolute zero implies zero kinetic energy).
James