We study a metric cubic gravity theory considering odd-parity modes of linear inhomogeneous perturbations on a spatially homogeneous Bianchi type I manifold close to the isotropic de Sitter spacetime. We show that in the regime of small anisotropy, the theory possesses new degrees of freedom compared to General Relativity, whose kinetic energy vanishes in the limit of exact isotropy. From the mass dispersion relation we show that such theory always possesses at least one ghost mode as well as a very short-time-scale (compared to the Hubble time) classical tachyonic (or ghost-tachyonic) instability. In order to confirm our analytic analysis, we also solve the equations of motion numerically and we find that this instability is developed well before a single e-fold of the scale factor. This shows that this gravity theory, as it is, cannot be used to construct viable cosmological models.
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