When a PN diode junction forward conducts heat is generated due to the product of forward volts and current.Is the heating evenly distributed between the P and N junctions ? or is it the P? or is it the N?.What is the story with a shottky does it go to the metal? Is this why common cathode is more common ? .
The PN junctions in a diode are so small that it would be difficult for any temperature gradient to exist. Also realize that a silicon PN is on one side 99.9999% silicon and something else and the other side is 99.9999% silicon and something else as well. So basically its mainly silicon. The P and N doping does not change the properties of the silicon that much.
For a schottky diode there would only be a difference if the heat resistance of the metal and silicon are very different which I doubt they are. Silicon is also a metal !
How is common Cathode more common ? This might have nothing to do with the properties of the diodes but more with the way that they are used in circuits.
I'd guess the heat is generated in the depletion region, between n and p. Which side has more depends on the doping level of each region. For the Schottky diode the depletion region is all in the semiconductor, so that's where heat is generated. The whole thing will thermalize (come to the same tempeature) fairly quickly.