  We derive the scalar potential in four spacetime dimensions from an
eight-dimensional $(R+\gamma R^4-2\Lambda-F_4^2)$ gravity model in the presence
of the 4-form $F_4$, with the (modified gravity) coupling constant $\gamma$ and
the cosmological constant $\Lambda$, by using the flux compactification of four
extra dimensions on a 4-sphere with the warp factor. The scalar potential
depends upon two scalar fields: the scalaron and the 4-sphere volume modulus.
We demonstrate that it gives rise to a viable description of cosmological
inflation in the early Universe, with the scalaron playing the role of inflaton
and the volume modulus to be (almost) stabilized at its minimum. We also
speculate about a possibility of embedding our model in eight dimensions into a
modified eight-dimensional supergavity that, in its turn, arises from a
modified eleven-dimensional supergravity.
