  This paper focuses on the cohomology of operator algebras associated with the
free semigroup generated by the set $\{z_{\alpha}\}_{\alpha\in\Lambda}$, with
the left regular free semigroup algebra $\mathfrak{L}_{\Lambda}$ and the
non-commutative disc algebra $\mathfrak{A}_{\Lambda}$ serving as two typical
examples. We establish that all derivations of these algebras are automatically
continuous. By introducing a novel computational approach, we demonstrate that
the first Hochschild cohomology group of $\mathfrak{A}_{\Lambda}$ with
coefficients in $\mathfrak{L}_{\Lambda}$ is zero. Utilizing the Ces\`aro
operators and conditional expectations, we show that the first normal
cohomology group of $\mathfrak{L}_{\Lambda}$ is trivial. Finally, we prove that
the higher cohomology groups of the non-commutative disc algebras with
coefficients in the complex field vanish when $|\Lambda|<\infty$. These methods
extend to compute the cohomology groups of a specific class of operator
algebras generated by the left regular representations of cancellative
semigroups, which notably include Thompson's semigroup.
