  Recent studies of quantum field theory in FLRW spacetime suggest that the
cause of the speeding up of the universe is the running vacuum (RV).
Appropriate renormalization of the energy-momentum tensor shows that the vacuum
energy density is a smooth function of the Hubble rate and its derivatives:
$\rho_{\rm vac}=\rho_{\rm vac}(H, \dot{H},\ddot{H},...)$. This is because in
QFT the quantum scaling of $\rho_{\rm vac}$ with the renormalization point
turns into cosmic evolution with $H$. As a result, any two nearby points of the
cosmic expansion during the standard FLRW epoch are smoothly related through
$\delta\rho_{\rm vac}\sim {\cal O}(H^2)$. In our approach, what we call the
`cosmological constant' $\Lambda$ is just the nearly sustained value of $8\pi
G(H)\rho_{\rm vac}(H)$ around (any) given epoch, where $G(H)$ is the running
gravitational coupling. In the present study, after summarizing the main QFT
calculations supporting the RV approach, we focus on the calculation of the
equation of state (EoS) of the RV for the entire cosmic history within such a
QFT framework. In particular, in the very early universe, where higher (even)
powers $\rho_{\rm vac}\sim{\cal O}(H^N)$ ($N=4,6,\dots$) triggered inflation
during a short period in which $H=$const, the vacuum EoS is very close to
$w_{\rm vac}=-1$. This ceases to be true during the FLRW era, where it adopts
the EoS of matter during the relativistic ($w_{\rm vac}=1/3$) and
non-relativistic ($w_{\rm vac}=0$) epochs. Interestingly enough, we find that
in the late universe the EoS becomes mildly dynamical and mimics quintessence,
$w_{\rm vac}\gtrsim-1$. It finally asymptotes to $-1$ in the remote future, but
in the transit the RV helps alleviating the $H_0$ and $\sigma_8$ tensions.
