In this work, we introduce the new cohomology groups depended on homomorphisms which are extensions of the topological Hochschild cohomology groups and investigate some of their properties that are analogue to the Hochschild cohomology groups. In addition, we use some homomorphisms on Banach algebras to define a new concept of amenability, namely, $(\sigma,\tau)$-super weak amenability which is a generalization of the cyclic amenability. Finally, we show that this new notion on a commutative Banach algebra ${\mathcal A}$ is equivalent to the $(\sigma,\tau)$-weak amenability, where $\sigma$ and $\tau$ are some continuous homomorphisms on ${\mathcal A}$.