Cohomology for the Lie algebra of type $A_2$ over a field of characteristic $2$

We calculate the cohomology of the classical Lie algebra of type $A_2$ over an algebraically closed field $k$ of characteristic $p=2$ with coefficients in simple modules. The obtained results were used to describe the cohomology of the Lie algebra $\mathfrak{gl} _3(k)$ and the cohomology of the restricted Lie algebra of Cartan type $W_3(\mathbf{1})$ with coefficients in the divided power algebra $O_3(\mathbf{1}).$