  Our Universe has an arrow of time. In accordance with the second law of
thermodynamics, entropy has been increasing ever since the Big Bang. The fact
that matter is in thermal equilibrium in the very early Universe, as indicated
by the cosmic microwave background, has led to the idea that gravitational
entropy must be very low in the beginning. Penrose proposed that gravitational
entropy can be quantified by the Weyl curvature, which increases as structures
formed. A concrete realization of such a measure is the Clifton-Ellis-Tavakol
gravitational entropy, which has been shown to be increasing in quite a number
of cosmological models. In this work, we show a counter-example involving a
class of inhomogeneous universes that are supported by a chameleon massless
scalar field and exhibit anisotropic spacetime shearing effects. In fact, in
our model the Clifton-Ellis-Tavakol gravitational entropy is increasing
although the magnitude of the Weyl curvature is decreasing; this is due to the
growth of the spacetime shear. The topology and the values of the three free
parameters of the model are constrained by imposing a positive energy density
for the cosmic fluid, and the thermodynamical requirements which follow from
the cosmological holographic principle and the second law. It is shown that a
negative deceleration parameter and a time decreasing Weyl curvature
automatically follow from those conditions. Thus, we argue that our model can
account for the formation of some primordial structures, like the Large Quasar
Groups, which has required a non-standard evolution of the spatial
anisotropies.
