  We study the behavior of the degeneration at the second step of the
Fr\"olicher spectral sequence of a $\mathcal{C}^{\infty}$ family of compact
complex manifolds. Using techniques from deformation theory and adapting them
to pseudo-differential operators we prove a result \textit{\`a la
Kodaira-Spencer} for the dimension of the second step of the Fr\"olicher
spectral sequence and we prove that, under a certain hypothesis, the
degeneration at the second step is an open property under small deformations of
the complex structure. We also provide an example in which we show that, if we
omit that hypothesis, we lose the openness of the degeneration.
