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In cubic-order Horndeski theories where a scalar field $\phi$ is coupled tononrelativistic matter with a field-dependent coupling $Q(\phi)$, we derive themost general Lagrangian having scaling solutions on the isotropic andhomogenous cosmological background. For constant $Q$ including the case ofvanishing coupling, the corresponding Lagrangian reduces to the form$L=Xg_2(Y)-g_3(Y)\square \phi$, where$X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$ and $g_2, g_3$ are arbitraryfunctions of $Y=Xe^{\lambda \phi}$ with constant $\lambda$. We obtain the fixedpoints of the scaling Lagrangian for constant $Q$ and show that the$\phi$-matter-dominated-epoch ($\phi$MDE) is present for the cubic coupling$g_3(Y)$ containing inverse power-law functions of $Y$. The stability analysisaround the fixed points indicates that the $\phi$MDE can be followed by astable critical point responsible for the cosmic acceleration. We propose aconcrete dark energy model allowing for such a cosmological sequence and showthat the ghost and Laplacian instabilities can be avoided even in the presenceof the cubic coupling.

Most general cubic-order Horndeski Lagrangian allowing for scaling solutions and the application to dark energy