We use recent measurements of the dimensionless coordinate distances from Fanaroff-Riley type IIb radio galaxies and the X-ray gas mass fractions in clusters to constrain the parameters of a nonstandard cosmological model. This work complements our recent analysis of the Type Ia supernova data within a non-Riemannian cosmological model. We use two independent data sets to constrain the new density parameter Ωψ, which is related to the non-Riemannian structure of the underlying spacetime and supplements the field equations that are very similar to the usual Friedmann equations of general relativity. Therefore, we place an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. The FR IIb and X-ray cluster data limit Ωψ0 to less than 10% of the critical density. The numerical results of this work also apply to several anisotropic cosmological models that, on the level of the field equations, exhibit a similar scaling behavior of the density parameters to that of our non-Riemannian model.