The gravitational energy-momentum pseudotensor: the cases of $f(R)$ and $f(T)$ gravity

We derive the gravitational energy-momentum pseudotensor $ \tau^{\sigma}_ {\phantom {\sigma} \lambda} $inmetric$ f\left (R \right) $ gravity and in teleparallel $f\left(T\right)$ gravity. In the first case, $R$ is the Ricci curvature scalar for a torsionless Levi-Civita connection; in the second case, $T$ is the curvature-free torsion scalar derived by tetrads and Weitzenb"ock connection. For both classes of theories the continuity equations are obtained in presence of matter. $f\left(R\right)$ and $f\left(T\right)$ are non-equivalent but differ for a quantity $\omega (T,B)$ containing the torsion scalar $T$ and a boundary term $B$. It is possible to obtain the field equations for $\omega (T,B)$ and the related gravitational energy-momentum pseudotensor $ \tau^{\sigma}_{\phantom {\sigma}\lambda} \vert \omega $. Finally we show that, thanks to this further pseudotensor, it is possible to pass from $f\left(R\right)$ to $ f \left ( T \right) $ and viceversa through a simple relation between gravitational pseudotensors.