In this brief note, we extend the initial investigation proposed in John et al. (2016) to study the Stokes equations with variable viscosity considering a stabilized finite element scheme based on equal order polynomials for approximating the velocity and the pressure. We establish optimal a priori error estimates for this new stabilized scheme, which are further confirmed by numerical tests. Moreover, these tests show a weak viscosity dependence in the error bounds for different orders of magnitude between 𝜈max and 𝜈min.