The super-weak model is a particle physics model which extends the Standard Model (SM) by a new U(1) gauge symmetry. The goal of this model is to investigate whether it is possible to have a non-vanishing parameter space capable of explaining multiple experimental results which require beyond SM physics. The extended particle spectrum includes the new mediator Z', a scalar singlet to deal with the meta-stability of the SM vacuum, and three right-handed neutrinos to account for the non-vanishing neutrino masses. In this talk, we explore the cosmological implications of such an extension. Our main focus will be on dark matter production and leptogenesis. In particular we find that a light -- mass of O(10) MeV -- sterile neutrino can play the role of dark matter via the resonant freeze-out scenario. We investigate present and possible future experimental bounds on the model parameters, both from particle physics experiments as well as from astrophysical observations.