  We explicitly compute the dynamics of closed homogeneous and isotropic
universes permeated by a single perfect fluid with a constant equation of state
parameter $w$ in the context of a recent reformulation of general relativity,
proposed in [1], which prevents the vacuum energy from acting as a
gravitational source. This is done using an iterative algorithm, taking as an
initial guess the background cosmological evolution obtained using standard
general relativity in the absence of a cosmological constant. We show that, in
general, the impact of the vacuum energy sequestering mechanism on the dynamics
of the universe is significant, except for the $w=1/3$ case where the results
are identical to those obtained in the context of general relativity with a
null cosmological constant. We also show that there are well behaved models in
general relativity that do not have a well behaved counterpart in the vacuum
energy sequestering paradigm studied in this paper, highlighting the specific
case of a quintessence scalar field with a linear potential.
