  Freudenthal duality in N=2, D=4 ungauged supergravity is generated by an
anti-involutive operator that acts on the electromagnetic fluxes, and results
to be a symmetry of the Bekenstein-Hawking entropy. We show that, with a
suitable extension, this duality can be generalized to the abelian gauged case
as well, even in presence of hypermultiplets. By defining Freudenthal duality
along the scalar flow, one can prove that two configurations of charges and
gaugings linked by the Freudenthal operator share the same set of values of the
scalar fields at the black hole horizon. Consequently, Freudenthal duality is
promoted to a nonlinear symmetry of the black hole entropy. We explicitly show
this invariance for the model with prepotential $F=-i X^0 X^1$ and
Fayet-Iliopoulos gauging.
