  There is a well-established connection between the higher-derivative
corrections to the black hole entropy and the black hole extremality bound. The
particular combination of EFT coefficients $c_i$ that controls the mass shift
at fixed charge and temperature also controls the entropy shift at fixed charge
and mass in the limit where the mass approaches the uncorrected value for the
extremal mass. In this note, we use the classical entropy function formalism to
examine the entropy corrections at exactly zero temperature, or at the
corrected value for the extremal mass. We find that the zero-temperature
entropy shift (a) is unrelated to the mass shift, (b) is $\mathcal{O}(c_i)$ in
the EFT coefficients, rather than $\mathcal{O}(\sqrt{c_i})$, as is the
constant-mass entropy shift, and (c) is negative in the example of the EFT
arising at low energies from QED plus gravity.
