  In this paper we study the heat equation (of Hodge-Laplacian) deformation of
$(p, p)$-forms on a K\"ahler manifold. After identifying the condition and
establishing that the positivity of a $(p, p)$-form solution is preserved under
such an invariant condition we prove the sharp differential Harnack (in the
sense of Li-Yau-Hamilton) estimates for the positive solutions of the
Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with
the K\"ahler-Ricci flow and some interpolating matrix differential Harnack type
estimates for both the K\"ahler-Ricci flow and the Ricci flow.
