Given Banach algebras $A$ and $B$ and $\theta\in\Delta(B)$. We shall study the Johnson pseudo-contractibility and pseudo-amenability of the $\theta$-Lau product $A\times_{\theta}B$. We show that if $A\times_{\theta}B$ is Johnson pseudo-contractible, then both $A$ and $B$ are Johnson pseudo-contractible and $A$ has a bounded approximate identity. In some particular cases, a complete characterization of Johnson pseudo-contractibility of $A\times_{\theta}B$ is given. Also, we show that pseudo-amenability of $A\times_{\theta}B$ implies the approximate amenability of $A$ and pseudo-amenability of $B$.