  Recently, Dinew and Popovici introduced and studied an energy functional $F$
acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic
metric and showed that in dimension 3 its critical points (if any) are
K\"ahler. In this article we further investigate the critical points of this
functional in higher dimensions and under holomorphic deformations. We first
prove that being a critical point for $F$ is a closed property under
holomorphic deformations. We then show that the existence of a K\"ahler metric
$\omega_k$ in the Aeppli cohomology class is an open property under holomorphic
deformations. Furthermore, we consider the case when the $(2,\,0)$-torsion form
$\rho_{\omega} ^{2,\,0}$ of $\omega$ is $\partial$-exact and prove that this
property is closed under holomorphic deformations. Finally, we give an explicit
formula for the differential of $F$ when the $(2,\,0)$-torsion form
$\rho_{\omega} ^{2,\,0}$ is $\partial$-exact.
