I have seen a number of posts here regarding the use of aluminum heat sinks. I have read only a few of them. For anyone seriously interested in the topic, they should read up on the Wiedemann-Franz law. A good place to start is the Wikipedia article on the Wiedemann-Franz law.
This law states that there is a temperature dependent ratio of thermal to electrical conductivities that does not depend on the metal. As typical for such "laws" they are only approximately correct. The interesting physics is in the deviations from Wiedemann-Franz law. This law seems to work well for good electrical conductors. The law is based upon the assumption that energy in a metal is carried primarily by electrons. Obviously, even excellent electrical insulators can conduct heat.
For good conductors such as aluminum, one should look at what limits electrical conductivity. Some of the limitations that limit conductivity in silver, copper, and aluminum are lattice defects. These include impurities and dislocations. Thus, best conductivity will be achieved with pure and annealed materials.
The other main difference will be in the interactions of electrons with phonons or lattice vibrations. That will give the main contribution to the Wiedemann-Franz ratio.
I do not wish to create a discourse on the subject. These are available. I just point out the situation.
Bill