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In this paper, we calculate cohomology of a classical Lie algebra of type$A_2$ over an algebraically field $k$ of characteristic $p=3$ with coefficientsin simple modules. To describe their structure, we will consider them asmodules over an algebraic group $SL_3(k).$ In the case of characteristic $p=3,$there are only two peculiar simple modules: a simple module isomorphic to thequotient module of the adjoint module by the center, and a one-dimensionaltrivial module. The results on the cohomology of simple nontrivial module areused for calculate the coomology of the adjoint module. We also calculatecohomology of the simple quotient algebra Lie of $A_2$ by the center.

Cohomology of simple modules for ${\mathfrak{sl}}_3(k)$ in characteristic $3$