That's right, here log(k) - H(p) is really the relative entropy (or KL divergence) between p and the uniform distribution, and all the same stuff is true for a

That's right, here log(k) - H(p) is really the relative entropy (or KL divergence) between p and the uniform distribution, and all the same stuff is true for a different "reference distribution" of the probabilities of balls landing in each bin. For discrete distributions the "absolute entropy" (just sum of -p log(p) as it shows up in Shannon entropy or statistical mechanics) is in this way really a special case of relative entropy. For continuous distributions, say over real numbers, the analogous quantity (integral of -p log(p)) isn't a relative entropy since there's no "uniform distribution over all real numbers". This still plays an important role in various situations and calculations...but, at least to my mind, it's a formally similar but conceptually separate object.