Noether and Belinfante corrected types of currents for perturbations in the Einstein-Gauss-Bonnet gravity

In the framework of an arbitrary $D$-dimensional metric theory, perturbations are considered on arbitrary backgrounds that are however solutions of the theory. Conserved currents for perturbations are presented following two known prescriptions: canonical N{\oe}ther theorem and Belinfante symmetrization rule. Using generalized formulae, currents in the Einstein-Gauss-Bonnet (EGB) gravity for arbitrary types of perturbations on arbitrary curved backgrounds (not only vacuum) are constructed in an explicit covariant form. Special attention is paid to the energy-momentum tensors for perturbations which are an important part in the structure of the currents. We use the derived expressions for two applied calculations: a) to present the energy density for weak flat gravitational waves in $D$-dimensional EGB gravity; b) to construct the mass flux for the Maeda-Dadhich-Molina 3D radiating black holes of a Kaluza-Klein type in 6D EGB gravity.