  Here, we investigate the growth of matter density perturbations as well as
the generalized second law (GSL) of thermodynamics in the framework of
$f(R)$-gravity. We consider a spatially flat FRW universe filled with the
pressureless matter and radiation which is enclosed by the dynamical apparent
horizon with the Hawking temperature. For some viable $f(R)$ models containing
the Starobinsky, Hu-Sawicki, Exponential, Tsujikawa and AB models, we first
explore numerically the evolution of some cosmological parameters like the
Hubble parameter, the Ricci scalar, the deceleration parameter, the density
parameters and the equation of state parameters. Then, we examine the validity
of GSL and obtain the growth factor of structure formation. We find that for
the aforementioned models, the GSL is satisfied from the early times to the
present epoch. But in the farther future, the GSL for the all models is
violated. Our numerical results also show that for the all models, the growth
factor for larger structures like the $\Lambda$CDM model fit the data very
well.
