EDITED
After reading the answers and many of the comments, I realised I did not put nearly enough restrictions on this question. The scenario I am assuming is as follows:
There are two cups of water of identical volume in cups that do not affect heat exchange (i.e. only the air temperature matters). One is at, say 80 C, and the other is at 5 C. The ambient air temperature is 25 C.
Obviously the warmer cup of water will cool faster than the colder cup, but my question is exactly why?
Again, I think it has to do with Newton's Law of Cooling and not much else. A friend of mine thinks it has more to do with thermodynamics, such as specific heat capacity varying at different temperatures.
Best Answer
Temperature difference is one thing. Newtons law of cooling will give that. However; there are many different methods for cooling and heating. In addition to conduction there is also heat transfer from convection (fluids) and radiation. Materials also have a thermal conductivity meaning different materials heat and cool differently. This conductivity is also temperature dependent; most notable at phase transitions (eg compare the conductivity of ice/water/steam - ice has higher thermal conductivity so this would counter your statement, but then there is also no convection, which is why cold water can feel colder than ice).
In a low radiant environment an object will generally have a net heat loss from radiation making the cold object warm slower than a warm onject cools. On a cold sunny day, the statement won't be true. Cold things will warm faster than hot things cool.