Local energy-momentum conservation in scalar-tensor-like gravity with generic curvature invariants

For a large class of scalar-tensor-like modified gravity whose action contains nonminimal couplings between a scalar field $\phi(x^\alpha)$ and generic curvature invariants $\mathcal{R}$ beyond the Ricci scalar $R=R^\alpha_{\;\;\alpha}$, we prove the covariant invariance of its field equation and confirm/prove the local energy-momentum conservation. These $\phi(x^\alpha)-\mathcal{R}$ coupling terms break the symmetry of diffeomorphism invariance under a particle transformation, which implies that the solutions of the field equation should satisfy the consistency condition $\mathcal{R}\equiv 0$ when $\phi(x^\alpha)$ is nondynamical and massless. Following this fact and based on the accelerated expansion of the observable Universe, we propose a primary test to check the viability of the modified gravity to be an effective dark energy, and a simplest example passing the test is the "Weyl/conformal dark energy".