  Sterile neutrinos with a mass in the eV range have been invoked as a possible
explanation of a variety of short baseline (SBL) neutrino oscillation
anomalies. However, if one considers neutrino oscillations between active and
sterile neutrinos, such neutrinos would have been fully thermalised in the
early universe, and would be therefore in strong conflict with cosmological
bounds. In this study we first update cosmological bounds on the mass and
energy density of eV-scale sterile neutrinos. We then perform an updated study
of a previously proposed model in which the sterile neutrino couples to a new
light pseudoscalar degree of freedom. Consistently with previous analyses, we
find that the model provides a good fit to all cosmological data and allows the
high value of $H_0$ measured in the local universe to be consistent with
measurements of the cosmic microwave background. However, new high $\ell$
polarisation data constrain the sterile neutrino mass to be less than
approximately 1 eV in this scenario. Finally, we combine the cosmological
bounds on the pseudoscalar model with a Bayesian inference analysis of SBL data
and conclude that only a sterile mass in narrow ranges around 1 eV remains
consistent with both cosmology and SBL data.
