  In General Relativity, there have been many proposals for defining the
gravitational energy density, notably those proposed by Einstein, Tolman,
Landau and Lifshitz, Papapetrou, M{\o}ller, and Weinberg. In this review, we
firstly explored the energy--momentum complex in an $n^{th}$ order
gravitational Lagrangian $L=L\left(g_{\mu\nu}, g_{\mu\nu,i_{1}},
g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots,
g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ and then in a gravitational
Lagrangian as \mbox{$L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p}
a_{k}R\Box^{k}R)\sqrt{-g}$}. Its gravitational part was obtained by invariance
of gravitational action under infinitesimal rigid translations using Noether's
theorem. We also showed that this tensor, in general, is not a covariant object
but only an affine object, that is, a pseudo-tensor. Therefore, the
pseudo-tensor $\tau^{\eta}_{\alpha}$ becomes the one introduced by Einstein if
we limit ourselves to General Relativity and its extended corrections have been
explicitly indicated. The same method was used to derive the energy--momentum
complex in $ f\left (R \right) $ gravity both in Palatini and metric
approaches. Moreover, in the weak field approximation the pseudo-tensor
$\tau^{\eta}_{\alpha}$ to lowest order in the metric perturbation $h$ was
calculated. As a practical application, the power per unit solid angle $\Omega$
emitted by a localized source carried by a gravitational wave in a direction
$\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge was
obtained, through the average value of the pseudo-tensor over a suitable
spacetime domain and the local conservation of the pseudo-tensor. As a
cosmological application, in a flat Friedmann--Lema\^itre--Robertson--Walker
spacetime, the gravitational and matter energy density in $f(R)$ gravity both
in Palatini and metric formalism was proposed.
