Despite the outstanding achievements of modern cosmology, the classical dispute on the precise value of $H_0$, which is the first ever parameter of modern cosmology and one of the prime parameters in the field, still goes on and on after over half a century of measurements. Recently the dispute came to the spotlight with renewed strength owing to the significant tension (at $>3σ$ c.l.) between the latest Planck determination obtained from the CMB anisotropies and the local (distance ladder) measurement from the Hubble Space Telescope (HST), based on Cepheids. In this work, we investigate the impact of the running vacuum model (RVM) and related models on such a controversy. For the RVM, the vacuum energy density $ρ_Λ$ carries a mild dependence on the cosmic expansion rate, i.e. $ρ_Λ(H)$, which allows to ameliorate the fit quality to the overall $SNIa+BAO+H(z)+LSS+CMB$ cosmological observations as compared to the concordance $Λ$CDM model. By letting the RVM to deviate from the vacuum option, the equation of state $w=-1$ continues to be favored by the overall fit. Vacuum dynamics also predicts the following: i) the CMB range of values for $H_0$ is more favored than the local ones, and ii) smaller values for $σ_8(0)$. As a result, a better account for the LSS structure formation data is achieved as compared to the $Λ$CDM, which is based on a rigid (i.e. non-dynamical) $Λ$ term.

Accepted for publication in Phys. Lett. B. Significantly extended discussion, two figures and references added