Why is an apple pies sauce always hotter than the pastry even though they
have been cooked on the same heat?
The paradox you describe arises from the ambiguity of the word 'hotter.' As you point out, the pastry and filling have been resting in the same oven for long enough to come to the same temperature - and therefore one is not hotter than the other in this sense. The molecules in the pastry and filling have the same average kinetic energy as the pie comes out of the oven.
Despite the temperatures being equal, your tongue is still more likely to get burned by the filling than the crust, though. There are 2 principles behind this: thermal conductivity and specific heat capacity.
Thermal conductivity is just the measure of how quickly heat energy travels through a substance. The pastry contains many pockets of air and cannot transport energy from a few microns away from your tongue to the interface with your tongue efficiently. Thus, as the outermost layer comes to equilibrium with your relatively cool mouth, more heat has a hard time rushing to the surface and flowing into your flesh where your nerves can sense it.
Specific heat capacity is something like energy density of a substance, and measures how much energy must be contained in a substance for it to have a certain temperature. For example, 100 grams of aluminum at 100 degrees C has more heat in it than 100 grams of copper at the same temperature. If you dropped both pieces of metal into separate cups of water, the one with the aluminum chunk will get warmer than the other- there's just more energy contained in it. Since the filling is mostly made of water, and water has a very high specific heat, the filling must give off a lot of heat for its temperature to decrease. This has 2 effects: when the pie comes out of the oven, the filling cools down much more slowly, and as a fragment of filling gives up heat to your tongue, it only cools down a tiny bit.
The long and short of it is- stick to ice cream- it's safer!
Rob landolfi, Science Teacher, Washington, DC
'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'