  Local higher-derivative corrections to the Einstein-Hilbert action yield
sub-leading corrections to the Bekenstein-Hawking area law. Here we show that
if the quantum effective action comprises a certain class of infrared
non-localities, the entropy of large black holes generally diverges to either
positive or negative infinity. In such theories, large spherically symmetric
black holes would be either highly chaotic or thermodynamically impossible,
respectively. In turn, this puts strong constraints on the Laurent expansion of
the form factors in the effective action.
