  Kaniadakis entropy is a one-parameter deformation of the classical
Boltzmann-Gibbs-Shannon entropy, arising from a self-consistent relativistic
statistical theory. Assuming a Kaniadakis-type generalization of the entropy
associated with the apparent horizon of Friedmann-Robertson-Walker (FRW)
Universe and using the gravity-thermodynamics conjecture, a new cosmological
scenario is obtained based on the modified Friedmann equations. By employing
such modified equations, we analyze the slow-roll inflation, driven by a scalar
field with power-law potential, at the early stages of the Universe. We explore
the phenomenological consistency of this model by computation of the scalar
spectral index and tensor-to-scalar ratio. Comparison with the latest Planck
data allows us to constrain Kaniadakis parameter to
$\kappa\lesssim\mathcal{O}(10^{-12}\div10^{-11})$, which is discussed in
relation to other observational bounds in the past literature. We also disclose
the effects of Kaniadakis correction term on the growth of perturbations at the
early stages of the Universe by employing the spherically symmetric collapse
formalism in the linear regime of density perturbations. We find out that the
profile of density contrast is non-trivially affected in this scenario.
Interestingly enough, we observe that increasing Kaniadakis parameter $\kappa$
corresponds to a faster growth of perturbations in a Universe governed by the
corrected Friedmann equations. Finally, we comment on the consistency of the
primordial power spectrum for scalar perturbations with the best data-fit
provided by Planck.
