We present a precise analysis to test hypothetical models involving sterile neutrinos beyond the standard flat-ΛCDM cosmology with the CMB observations from the Planck mission and BAO measurements. This analysis shows that adding the locally measured Hubble parameter H 0 = 73.00 ± 1.75 km s −1 Mpc −1 to the data removes the need for the informative physical m thermal sterile prior in CMB constraints of m ef f ν,sterile . Under the constraints from the data containing the locally measured H 0 we obtain an upper limit m ef f ν,sterile < 0.306 eV scale mass for the massive sterile neutrino, and an upper limit Σm ν < 0.214 eV scale mass for the three degenerate massive neutrino (95 per cent confidence level). We also obtain the value σ 8 = 0.81 +0.05 −0.06 (95 per cent confidence level), which is in compatibility with the constraints from Planck 2015 CMB data at the 1σ level. We find that introducing parameter m ef f ν,sterile to the model of cosmology reduces the σ 8 value and moves it closer to the obtained value for this parameter from the KiDS-450 analysis. Our results show that the locally measured Hubble parameter can increase constraints on σ 8 values.