Let $\mathcal A$ be a Banach algebra and $\sigma$ and $\tau$ be continuous endomorphisms on $\mathcal A$. In this paper, we investigate $(\sigma, \tau)-$amenability and $(\sigma, \tau)-$weak amenability for unitization of Banach algebras, and also the relation between of them. We introduce and study the concepts $(\sigma, \tau)-$trace extention property, $(\sigma, \tau)-I-$weak amenability and $(\sigma, \tau)-$ideal amenability for $\mathcal A$ and its unitization, where $I$ is a closed two-sided ideal in $\mathcal A$.
