  Let $X$ be a compact complex manifold and $E$ be a holomorphic vector bundle
on $X$. Given a deformation $(\mathcal{X},\mathcal{E})$ of the pair $(X,E)$
over a small polydisk $B$ centered at the origin, we study the jumping
phenomenon of the cohomology groups
$\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ near $t = 0$. Generalizing
previous results of X. Ye for the tangent bundle $E = T_{\mathcal{X}_t}$ and
exterior powers of the cotangent bundle $E = \Omega^p_{\mathcal{X}_t}$, we show
that there are precisely two cohomological obstructions to the stability of
$\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$, which can be expressed
explicitly in terms of the Maurer-Cartan element associated to the deformation
$(\mathcal{X},\mathcal{E})$. As an application, we study the jumping phenomenon
of the dimension of the cohomology group
$H^1(\mathcal{X}_t,\text{End}(T_{\mathcal{X}_t}))$ which is related to a
question raised by physicists.
