With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced a notion of visible action for holomorphic actions of Lie groups on complex manifolds. His propagation theorem of the multiplicity-freeness property produces various kinds of multiplicity-free theorems for unitary representations realized in the space of holomorphic sections of an equivariant holomorphic vector bundle whose base space admits a visible action of a Lie group. In this article, we consider the cohomology space of a differential complex of a specific type on a real analytic manifold following papers of Schmid and Wong, and obtain the multiplicity-freeness property of the cohomology space under certain conditions. We also show applications of the main result to the Dolbeault cohomology space of a homogeneous line bundle on an elliptic orbit of a real linear semisimple Lie group.