According to Lovelock ’ s theorem , the Hilbert-Einstein and the Lovelock actions are indistinguishable from their field equations .
However , they have different scalar-tensor counterparts , which correspond to the Brans-Dicke and the Lovelock-Brans-Dicke ( LBD ) gravities , respectively .
In this paper the LBD model of alternative gravity with the Lagrangian density \mathscr { L } _ { \text { LBD } } = \frac { 1 } { 16 \pi } \left [ \phiup \left ( R + \frac { a } { \sqrt { - g } % } { } ^ { * } RR + b \mathcal { G } \right ) - \frac { \omega _ { \text { L } } } { \phiup } \nabla _ { \alpha } % \phiup \nabla ^ { \alpha } \phiup \right ] is developed , where ^ { * } RR and \mathcal { G } respectively denote the topological Chern-Pontryagin and Gauss-Bonnet invariants .
The field equation , the kinematical and dynamical wave equations , and the constraint from energy-momentum conservation are all derived .
It is shown that , the LBD gravity reduces to general relativity in the limit \omega _ { \text { L } } \to \infty unless the “ topological balance condition ” holds , and in vacuum it can be conformally transformed into the dynamical Chern-Simons gravity and the generalized Gauss-Bonnet dark energy with Horndeski-like or Galileon-like kinetics .
Moreover , the LBD gravity allows for the late-time cosmic acceleration without dark energy .
Finally , the LBD gravity is generalized into the Lovelock-scalar-tensor gravity , and its equivalence to fourth-order modified gravities is established .
It is also emphasized that the standard expressions for the contributions of generalized Gauss-Bonnet dependence can be further simplified .

Key words : Lovelock ’ s theorem , topological effects , modified gravity