  In Starobinsky inflation with a Weyl squared Lagrangian $-\alpha C^2$, where
$\alpha$ is a coupling constant, we study the linear stability of cosmological
perturbations on a spatially flat Friedmann-Lema\^{i}tre-Robertson-Walker
background. In this theory, there are two dynamical vector modes propagating as
ghosts for $\alpha>0$, whose condition is required to avoid tachyonic
instabilities of vector perturbations during inflation. The tensor sector has
four propagating degrees of freedom, among which two of them correspond to
ghost modes. However, the tensor perturbations approach constants after the
Hubble radius crossing during inflation, and hence the classical instabilities
are absent. In the scalar sector, the Weyl curvature gives rise to a ghost mode
coupled to the scalaron arising from the squared Ricci scalar. We show that two
gauge-invariant gravitational potentials, which are both dynamical in our
theory, are subject to exponential growth after the Hubble radius crossing.
There are particular gauge-invariant combinations like the curvature
perturbations whose growth is suppressed, but it is not possible to remove the
instability of other propagating degrees of freedom present in the perturbed
metric. This violent and purely classical instability present in the scalar
sector makes the background unviable. Furthermore, the presence of such
classical instability makes the quantization of the modes irrelevant, and the
homogeneous inflationary background is spoiled by the Weyl curvature term.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory