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For two Banach algebras $A$ and $B$, the $T$-Lau product $A\times_T B$, wasrecently introduced and studied for some bounded homomorphism $T:B\to A$ with$\|T\|\leq 1$. Here, we give general nessesary and sufficent conditions for$A\times_T B$ to be (approximately) cyclic amenable. In particular, we extendsome recent results on (approximate) cyclic amenability of direct product$A\oplus B$ and $T$-Lau product $A\times_T B$ and answer a question on cyclicamenability of $A\times_T B$.

More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism