In this paper we calculate the linear perturbations of the cosmological redshift drift. We show explicitly that our expressions are gauge-invariant and compute the power spectrum of the redshift drift perturbations and its correlations with galaxy number counts within linear perturbation theory. Our findings show that the perturbations are small, and that the peculiar velocity and acceleration terms are dominating and cannot be neglected when modeling the full perturbative expression for the redshift drift. We also find that the cross-correlations with galaxy number count fluctuations might increase the detectability of the effect and can help to separate the perturbative effects from the background cosmological redshift drift signal.
