Off-shell Noether current and conserved charge in Horndeski theory

We derive the off-shell Noether current and potential in the context of Horndeski theory, which is the most general scalar–tensor theory with a Lagrangian containing derivatives up to second order while yielding at most to second-order equations of motion in four dimensions. Then the formulation of conserved charges is proposed on basis of the off-shell Noether potential and the surface term got from the variation of the Lagrangian. As an application, we calculate the conserved charges of black holes in a scalar–tensor theory with non-minimal coupling between derivatives of the scalar field and the Einstein tensor