  We investigate the cosmic evolutions in the extended Starobinsky model (eSM)
obtained by adding one $R^{ab}R_{ab}$ term to the Starobinsky model. We discuss
the possibility of various cosmic evolutions with a special focus on the
radiation-dominated era (RDE). Using simple assumptions, a second-order
non-linear differential equation describing the various cosmic evolutions in
the eSM is introduced. By solving this non-linear equation numerically, we show
that the various cosmic evolutions, such as the standard cosmic evolution ($a
\propto t^{1/2}$) and a unique oscillating cosmic evolution, are feasible due
to the effects of higher-order terms introduced beyond Einstein's gravity.
Furthermore, we consider big bang nucleosynthesis (BBN), which is the most
important observational result in the RDE, to constrain the free parameters of
the eSM. The primordial abundances of the light elements, such as $^{4}$He, D,
$^{3}$He, $^{7}$Li, and $^{6}$Li by the cosmic evolutions are compared with the
most recent observational data. It turns out that most non-standard cosmic
evolutions can not easily satisfy these BBN constraints, but a free parameter
of the viable models with the oscillating cosmic evolution is shown to have an
upper limit by the constraints. In particular, we find that the free parameter
is most sensitive to deuterium and $^4$He abundances, which are being precisely
measured among other elements. Therefore, more accurate measurements in the
near future may enable us to distinguish the eSM from the standard model as
well as other models.
