  Motivated by the modified gravity theories $F(R)\neq R$ and inflationary
physics, we first propose and investigate an inflation model in a scaled
gravity $F(R)=R\,+\beta R$, where $\beta $ is a dimensionless scaling
parameter. The latter is also implemented in a particular potential $V(\phi
)=M^{4}\left[ 1-\cos \left( \frac{\phi }{\mu}\right)^{\beta }\right] $ being
considered to drive the inflation via a parameter coupling scenario. Using the
slow-roll approximations, the gravity scale parameter $\beta $ is approached
with respect to the range of the associated computed cosmological observables
$n_{s}$ and $r$ according to the recent Planck and BICEP/Keck data. Then, we
discuss the axionic dark matter in the suggested gravity model by considering
the case where the inflaton is taken to be identified with an axion-like field
$\phi =f_{a}\theta $ with the decay constant $f_{a}=\mu$. Referring to the
known data, the underlying inflation scale $M$ is constrained to be much lower
than the corresponding axion scale $M\ll f_{a}$.
