  In this work we investigate the most general non-minimally coupled
$\mathbb{Z}_2$ symmetric scalar-tensor effective field theory (EFT) of gravity
up to dimension six in the operator expansion. The most general action is
presented along with its equations of motion both in the covariant form and
also in the coordinate form resulting from an FLRW analysis. The pressure and
density of the scalar field are found as well as the equation of state
parameter and some cosmological parameters. We analyse the background evolution
of the scalar field within the framework of the slow-roll approximation and
provide a brief discussion of $\mathbb{Z}_2$ symmetry breaking of the scalar
field which gives an order of magnitude constraint on one of the coupling
constants. To provide a concrete example we choose a couple of potentials for
the scalar field and explore the cosmology. Some brief comments on the range of
validity of the EFT are also offered as well as the connections between the
general model and well-known examples in the literature.
