  We compute the number density of massive Black Holes (BHs) at the centre of
galaxies at z=6 in different Dynamical Dark Energy (DDE) cosmologies, and
compare it with existing observational lower limits, to derive constraints on
the evolution of the Dark Energy equation of state parameter w. Our approach
only assumes the canonical scenario for structure formation from the collapse
of overdense regions of the Dark Matter dominated primordial density field on
progressively larger scales; the Black Hole accretion and merging rate have
been maximized in the computation so as to obtain robust constraints on w and
on its look-back time derivative w_a. Our results provide independent
constraints complementary to those obtained by combining Supernovae, Cosmic
Microwave Background and Baryonic Acoustic Oscillations; while the latter
concern combinations of w_0 and w_a leaving the time evolution of the state
parameter w_a highly unconstrained, the BH abundance mainly provide upper
limits on w_a, only weakly depending on w_0. Combined with the existing
constraints, our results significantly restrict the allowed region in DDE
parameter space, ruling out DDE models not providing cosmic time and fast
growth factor large enough to allow for the building up of the observed
abundance of BHs; in particular, models with -1.2 \leq w_0 \leq -1 and positive
redshift evolution w_a > 0.8 - completely consistent with previous constraints
- are strongly disfavoured by our independent constraints from BH abundance.
Such range of parameters corresponds to "Quintom" DDE models, with w crossing
-1 starting from larger values.
