A hypercube's surface area to volume ratio grows without bound with dimension, so radiating heat ought to be easier with more dimensions even if you build a fairly blobby (I'm sure that's the technical term) core, though an n-sphere's peaks at n=7, so you don't want to get too blobby in higher dimensions.
Took me a minute to figure out what you are saying. Then I realized you are referring to the 16pi^3/15 surface area ratio of a 7-sphere. After n>7 it does seems to infinitely move towards zero. But the heat equasion is different in higher dimensional space and governed by the parabolic Harnack inequalities. I'm not sure how surface area makes sense here.
