  We consider the de Rham complex over scales of weighted isotropic and
anisotropic H\"older spaces with prescribed asymptotic behaviour at the
infinity. Starting from theorems on the solvability of the system of operator
equations generated by the de Rham differential $d$ and the operator $d^*$
formally adjoint to it, a description of the cohomology groups of the de Rham
complex over these scales was obtained. It was also proved that in the
isotropic case the cohomology space is finite-dimensional, and in the
anisotropic case the general form of an element from the cohomology space is
presented.
