  We show that the physical conditions which induce the Thakurta metric,
recently studied by B{\oe}hm {\it et al.} in the context of time-dependent
black hole masses, correspond to a single accreting black hole in the entire
Universe filled with isotropic non-interacting dust. In such a case, the
physics of black hole accretion is not local but tied to the properties of the
whole Universe. We show that radiation, primordial black holes or particle dark
matter cannot produce the specific energy flux required for supporting the mass
growth of Thakurta black holes. In particular, this solution does not apply to
black hole binaries. We conclude that cosmological black holes and their mass
growth cannot be described by the Thakurta metric, and thus existing
constraints on the primordial black hole abundance from the LIGO-Virgo and the
CMB measurements remain valid.
