Freudenthal duality, introduced in L. Borsten, D. Dahanayake, M. J. Duff and
W. Rubens, Phys.Rev. D80, 026003 (2009), and defined as an anti-involution on
the dyonic charge vector in d = 4 space-time dimensions for those dualities
admitting a quartic invariant, is proved to be a symmetry not only of the
classical Bekenstein-Hawking entropy but also of the critical points of the
black hole potential. Furthermore, Freudenthal duality is extended to any
generalized special geometry, thus encompassing all N > 2 supergravities, as
well as N = 2 generic special geometry, not necessarily having a coset space
structure.Comment: 1+9 pages; v2: some parts rewritten, Ref. added, to appear in PL
