  In the present paper we analyze a toy model for an $f(\phi,R)$ gravity which
has the form of a power-law modified gravity in which the exponent is
space-time dependent. Namely, we investigate the effects of adding to the
Hilbert-Einstein action an $R^{\phi}$-term. We present possible equivalences of
the model with known models of modified gravity theories and examine the
problem of matter stability in this model. Like $f(R)$-gravity toy models, the
present one offers the possibility of unifying the early and the late-time
evolution of the Universe. We show that the behavior of the scalar field
depends globally on the size of the Universe and locally on the surrounding
environment. For the early Universe it lets appear a huge cosmological constant
that might drive inflation. For the late-times it lets appear globally a tiny
cosmological constant.
