The $\Lambda$CDM cosmology, which includes a Cosmological Constant (CC), has been the dominant paradigm for the past 25 years. However, the appearance of tensions in different cosmological parameters and the persistence of traditional theoretical problems associated with the CC challenge the validity and viability of the model, leading to the search for new physics. Recent studies of vacuum energy in quantum field theory in a FLRW spacetime predict that this new physics may be related to a slow running of the vacuum energy density with the Hubble function. The running can be described by the difference of two close values near the present, $\delta\rho_{\rm vac}\sim \mathcal{O}(H^2)$, which is a characteristic of the traditional Running Vacuum Models. Higher powers of $H$ may be relevant in the early universe and could naturally drive a mechanism for inflation. On the other hand, the equation of state (EoS) of the running vacuum is no longer predicted to be fixed at exactly -1. Instead, it is expected to evolve, mimicking the dominant component at the time. Thus, during inflation, it starts at $w_{\rm vac}-1$, during the radiation-dominated epoch, $w_{\rm vac}=1/3$, during the matter-dominated epoch, $w_{\rm vac}=0$, and near the present, it behaves as either quintessence or phantom. The additional features revealed by these calculations may lead to a consistent model capable of alleviating the current $\sigma_8$ and $H_0$ tensions of modern cosmology and shedding light on the problem of the CC.