  We explore the influences of the higher order Gauss Bonnet (GB) correction
terms on the growth of perturbations at the early stage of a (n +
1)-dimensional Friedmann-Robertson-Walker (FRW) universe. Considering a
cosmological constant in the FRW background, we study the linear perturbations
by adopting the spherically symmetric collapse (SC) formalism. In light of the
modifications that appear in the field equations, we disclose the role of the
GB coupling constant {\alpha}, as well as the extra dimensions n > 3 on the
growth of perturbations. It, in essence, is done by defining a dimensionless
parameter \{beta}=(n-2)(n-3) \{alpha} H_0^2 in which H_0 is the Hubble
constant. We find that the matter density contrast starts growing at the early
stages of the universe and, as the universe expands, it grows faster compared
to the standard cosmology. Besides, in the framework of GB gravity, the growth
of matter perturbations in higher dimensions is faster than its standard
counterpart (n = 3). Further, in the presence of {\alpha}, the growth of
perturbations increases as it increases. This is an expected result, since the
higher order GB correction terms increase the strength of the gravity and thus
support the growth of perturbations. For the existing cosmological model, we
also investigate the behavior of quantities such as density abundance,
deceleration, and the jerk parameter. Finally, we study the imprint of the GB
parameter and the higher dimensions in the evolution of the mass function of
the dark matter halos.
