For two Banach algebras $ A$ and $ B$ and a non-zero multiplicative linear functional $\theta$ on $ B$, Monfared introduced the $\theta-$Lau product structure $A\times_{\theta}B$. In this paper, we investigate and study the notions of $\phi-$biprojectivity, $\phi-$biflatness and $\phi-$Johnson amenability of $A\times_{\theta} B$ and their relation with $ A$ and $ B$. As an application, we characterize $\phi$-biflatness, $\phi$-biprojectivity and $\phi$-Johnson amenability for $\theta-$Lau product of Banach algebras related to locally compact groups and discrete semigroups
