  We apply the gravity-thermodynamics conjecture, namely the first law of
thermodynamics on the Universe horizon, but using the generalized Kaniadakis
entropy instead of the standard Bekenstein-Hawking one. The former is a
one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy,
arising from a coherent and self-consistent relativistic statistical theory. We
obtain new modified cosmological scenarios, namely modified Friedmann
equations, which contain new extra terms that constitute an effective dark
energy sector depending on the single model Kaniadakis parameter $K$. We
investigate the cosmological evolution, by extracting analytical expressions
for the dark energy density and equation-of-state parameters and we show that
the Universe exhibits the usual thermal history, with a transition redshift
from deceleration to acceleration at around 0.6. Furthermore, depending on the
value of $K$, the dark energy equation-of-state parameter deviates from
$\Lambda$CDM cosmology at small redshifts, while lying always in the phantom
regime, and at asymptotically large times the Universe always results in a
dark-energy dominated, de Sitter phase. Finally, even in the case where we do
not consider an explicit cosmological constant the resulting cosmology is very
interesting and in agreement with the observed behavior.
