We investigate the cosmological applications of new gravitational scalar-tensor
theories and we analyze them in the light of $H_0$ tension. In these theories
the Lagrangian contains the Ricci scalar and its first and second derivatives
in a specific combination that makes them free of ghosts, thus corresponding
to healthy bi-scalar extensions of general relativity.
We examine two specific models, and for particular choices of the model
parameters we find that the effect of the additional terms is negligible at
high redshifts, obtaining a coincidence with $\Lambda$CDM cosmology, however
as time passes the deviation increases and thus at low redshifts the Hubble
parameter acquires increased values ($H_0\approx 74 km/s/Mpc$) in a controlled
way. The mechanism behind
this behavior is the fact that the effective dark-energy equation-of-state
parameter exhibits phantom behavior, which implies
faster expansion, which is one of the theoretical requirements that are
capable of alleviating the $H_0$ tension.
Lastly, we
confront the models with Cosmic Chronometer (CC) data showing full
agreement within 1$\sigma$ confidence level.