  We consider the ghost-free higher order corrections to the Starobinsky model
in the old minimal supergravity. In general, higher order corrections cannot be
forbidden by symmetries, which likely violate the flatness of the scalaron
potential and makes inflation difficult to explain the present Universe. We
find a severe constraint on the dimensionless coupling of the $R^4$ correction
as $-5.5 \times 10^{-8}<s<9.1 \times 10^{-8}$ from the recent results of the
Planck observation. If we start from the chaotic initial condition, the
constraint becomes much severer. However, in the case where the coupling of the
$R^4$ correction is positive, the scalaron potential has a local maximum with
two local minimum at the origin and infinity, which admits topological
inflation. In this case, inflation can take place naturally if the coupling
satisfies the observational constraints.
