  We investigate the cosmological applications of new gravitational
scalar-tensor theories, which are novel modifications of gravity possessing 2+2
propagating degrees of freedom, arising from a Lagrangian that includes the
Ricci scalar and its first and second derivatives. Extracting the field
equations we obtain an effective dark energy sector that consists of both extra
scalar degrees of freedom, and we determine various observables. We analyze two
specific models and we obtain a cosmological behavior in agreement with
observations, i.e. transition from matter to dark energy era, with the onset of
cosmic acceleration. Additionally, for a particular range of the model
parameters, the equation-of-state parameter of the effective dark energy sector
can exhibit the phantom-divide crossing. These features reveal the capabilities
of these theories, since they arise solely from the novel, higher-derivative
terms.
