M. Carmen Minguez in this article constructs a homomorphism between de Rham and Cubical Singular Cohomology without showing that is an isomorphism. This is done in the context of Synthetic Differential Geometry.

In general Synthetic Differential geometers seem to be quite aware of the cubical setup, probably because differential forms naturally get a cubical structure if defined via infinitesimals.

  Moerdijk/Reyes, however, in their book on synthetic differential geometry choose to establish the isomorphism between de Rham and simplicial singular cohomology using infinitesimals (p.155 and on)

M. Carmen Minguez in this article constructs a homomorphism between de Rham and Cubical Singular Cohomology without showing that is an isomorphism. This is done in the context of Synthetic Differential Geometry.

In general Synthetic Differential geometers seem to be quite aware of the cubical setup, probably because differential forms naturally get a cubical structure if defined via infinitesimals. This is very visible in chapter IV, section 1 of Moerdijk/Reyes' book on synthetic differential geometry. In remark 1.8 (p. 145) they mention a bridge to cubical (co)homology - then, however, they choose to establish the isomorphism between de Rham and simplicial singular cohomology.