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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 BH 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 ≤ w  0 ≤−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.

Constraining dynamical dark energy models through the abundance of high‐redshift supermassive black holes