  We compute the Chern-Weil map introduced by Hajac and Maszczyk associated
with a Hopf-Galois extension in the case of $2$-cocycle deformation of the
latter. We analyze the cases where the $2$-cocycle belongs to the structure
Hopf algebra, an external symmetry, and the combined case. Moreover, we prove
the naturality of the Chern-Weil map both in the deformed and not deformed case
generalizing the classical result to the non-commutative case.
