  We construct black holes with scalar hair in a wide class of four-dimensional
N=2 Fayet-Iliopoulos gauged supergravity theories that are characterized by a
prepotential containing one free parameter. Considering the truncated model in
which only a single real scalar survives, the theory is reduced to an
Einstein-scalar system with a potential, which admits at most two AdS critical
points and is expressed in terms of a real superpotential. Our solution is
static, admits maximally symmetric horizons, asymptotically tends to AdS space
corresponding to an extremum of the superpotential, but is disconnected from
the Schwarzschild-AdS family. The condition under which the spacetime admits an
event horizon is addressed for each horizon topology. It turns out that for
hyperbolic horizons the black holes can be extremal. In this case, the
near-horizon geometry is AdS_2 x H^2, where the scalar goes to the other,
non-supersymmetric, critical point of the potential. Our solution displays
fall-off behaviours different from the standard one, due to the fact that the
mass parameter $m^2=-2/\ell^2$ at the supersymmetric vacuum lies in a
characteristic range $m^2_{BF}\le m^2\le m^2_{\rm BF}+\ell^{-2}$ for which the
slowly decaying scalar field is also normalizable. Nevertheless, we identify a
well-defined mass for our spacetime, following the prescription of Hertog and
Maeda. Quite remarkably, the product of all horizon areas is not given in terms
of the asymptotic cosmological constant alone, as one would expect in absence
of electromagnetic charges and angular momentum. Our solution shows
qualitatively the same thermodynamic behaviour as the Schwarzschild-AdS black
hole, but the entropy is always smaller for a given mass and AdS curvature
radius. We also find that our spherical black holes are unstable against radial
perturbations.
