Rubbish! Although I have been told in no uncertain terms I have to dissociate this opinion from "martlou" who is my other half and believes this myth (and many others).
I will try and explain this from a physical (thermodynamics) point of view.
Let us say we have two cups of water, both exactly the same in every way except the temperature of the water. Cup 'C' has cold water initially at 5 degrees C while cup 'H' has hot water at temperature 55 degrees C.
Both are placed in a very cold environment, say outside tonight, which is about minus 10 degrees C.
Both cups of water will begin cooling. The RATE of cooling from cup 'H' will be initially be faster than cup 'C' because the temperature difference is greater, but the water in cup 'C' will freeze first. The rate of cooling of cup 'C' will be lower but it has only 15 degrees to fall whereas the other 'H' cup has 65 degrees C to fall.
This hasn't proved anything just yet but consider, what happens to the water in cup 'H' when it cools to 5 degrees C (i.e. the original temperature of the water in cup 'C'?). Has the water in cup 'C' just sat there waiting for the hot water to catch up?!!
Mon dieu - you will be telling me next that you unzip and take off your coat when you go inside so that you can put it back on and 'feel the benefit' when you go back outside.
Happy flaming guys and gals.
As an aside - may be it was a mistake to describe the containers as cups because now all I can think of is an odd looking bra.