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".
