  In this paper, we study the $F(R, G)$ gravity model with an interacting model
by flat-FRW metric in a viscous fluid. We consider that the universe dominates
with components of dark matter and dark energy. This means that the dark matter
component derives from Extended Bose-Einstein Condensate (EBEC) and the
components of dark energy arise from the $F(R, G)$ gravity. After obtaining the
Einstein equation, the energy density and the pressure of dark energy are
written in terms of the geometries of the curvature and the Gauss-Bonnet terms,
and components of dark matter and viscous fluid. Also, the corresponding
continuity equations are written with the presence of interaction terms. In
what follows, we employ the EBEC regime instead of the normal dark matter by
the dark matter Equation of State (EoS) as $p_{dm} = \alpha \rho_{dm} + \beta
\rho_{dm}^2$, which arises from the gravitational form. The EoS can be
expressed from the perspective of the virial expansion, in which the first and
second terms represent normal dark matter and quantum ground state. Next, the
corresponding Friedmann equations reconstruct in terms of the redshift
parameter, then by using the scenario of the power-law cosmology for the scale
factor, we fit the present model with the Hubble amounts of 51 supernova data
by the likelihood analysis. In that case, we acquire the cosmological
parameters of dark energy in terms of the redshift parameter, and by plotting
these graphs, we see that the universe is currently undergoing an accelerated
expansion phase. Finally, we investigate the stability of the present model
with the sound speed parameter.
