  In arXiv:1207.2679 a new prescription for finding nonextremal black hole
solutions to N=2, D=4 Fayet-Iliopoulos gauged supergravity was presented, and
explicit solutions of various models containing one vector multiplet were
constructed. Here we use the same method to find new nonextremal black holes to
more complicated models. We also provide a general recipe to construct non-BPS
extremal solutions for an arbitrary prepotential, as long as an axion-free
condition holds. These follow from a set of first-order conditions, and are
related to the corresponding supersymmetric black holes by a multiplication of
the charge vector with a constant field rotation matrix S. The fake
superpotential driving this first-order flow is nothing else than Hamilton's
characteristic function in a Hamilton-Jacobi formalism, and coincides in the
supersymmetric case (when S is plus or minus the identity) with the
superpotential proposed by Dall'Agata and Gnecchi in arXiv:1012.3756. For the
nonextremal black holes that asymptote to (magnetic) AdS, we compute both the
mass coming from holographic renormalization and the one appearing in the
superalgebra. The latter correctly vanishes in the BPS case, but also for
certain values of the parameters that do not correspond to any known
supersymmetric solution of N=2 gauged supergravity. We finally show that the
product of all horizon areas depends only on the charges and the asymptotic
value of the cosmological constant.
