Let $\mathcal A$ and $\mathcal B$ be two Banach algebras and $\theta$ be a nonzero character on $\mathcal B$. For two homomorphisms $\sigma$ and $\tau$ on Banach algebras $\mathcal A$ and $\mathcal B$ respectively, we deal with the relations between the $\sigma$-amenability of $\mathcal A$ and $\tau$-amenability of $\mathcal B$ with $(\sigma, \tau)-$amenability of their $\theta$-Lau product ${\mathcal A}\times_{\theta}{\mathcal B}$ under certain conditions. Moreover, we study the hereditary properties of $(\sigma, \tau)$-amenability of $\theta$-Lau product ${\mathcal A}\times_{\theta}{\mathcal B}$.
