For Banach algebra $\cal B$ and Banach $\cal B$-bimodule $\cal A$, the amalgamated of $\cal B$ along a Banach $\cal B$-bimodule $\cal A$, i.e. ${\cal A}\rtimes{\cal B}$, was introduce by Javanshiri and Nemati \cite{JN}. In this paper, for continuous endomorphisms $\sigma, \sigma'$ and $\tau, \tau'$ on $\cal A$ and $\cal B$ respectively, we investigate the $((\sigma, \tau), (\sigma', \tau'))-$amenability of ${\cal A}\rtimes{\cal B}$, and also we study the relations between of $((\sigma, \tau), (\sigma', \tau'))-$amenability of ${\cal A}\rtimes{\cal B}$ with similar concept on $\cal A$ and $\cal B$.
