  Johnson has shown [arXiv:1703.06119] that in the vicinity of a critical point
the efficiency of a black hole heat engine can approach the Carnot efficiency
while maintaining finite power. We characterize and extend this result in
several ways, and we show how the rate of approach to the Carnot efficiency is
governed by the critical exponents. We apply these results to several classes
of black holes to illustrate their validity. Odd-order Lovelock black holes are
known to have isolated critical points for which the critical exponents differ
from the mean field theory values, providing a non-trivial test of the results.
In this case, our results indicate the impossibility of even-order Lovelock
black holes with isolated critical points in this class: their existence would
constitute a violation of the second law of thermodynamics.
