Summary: There have been few examples of computations of Sweedler cohomology, or its generalization in low degrees known as lazy cohomology, for Hopf algebras of positive characteristic. In this paper we first provide a detailed calculation of the Sweedler cohomology of the algebra of functions on \((\mathbb{Z}/2)^r\), in all degrees, over a field of characteristic 2. Here the result is strikingly different from the characteristic zero analog.{ }Then we show that there is a variant in characteristic \(p\) of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group; in positive characteristic, the statement is, rather surprisingly, simpler.