  We propose a new formulation of $f(R)$ gravity, dubbed scalarized $f(R)$
gravity, in which the Legendre transform is included as a dynamical term. This
leads to a theory with second-order field equations that describes general
relativity with a self-interacting scalar field, without requiring the
introduction of conformal frames. We demonstrate that the quadratic version of
scalarized $f(R)$ gravity reduces to general relativity with a massive scalar
field, and we explore its implications for Friedmann cosmology. Our findings
suggest that scalarized $f(R)$ gravity may lead to simplified descriptions of
cosmological applications, while the proposed formulation could offer a new
perspective on the relationship between $f(R)$ gravity and scalar-tensor
theories.
