  We find the generalized version of the Toomre's criterion for the stability
of a rotating thin disk in the context of Eddington inspired Born-Infeld (EiBI)
gravity which possesses one free parameter $\chi$. To do so we use the weak
field limit of the theory and find the dispersion relation for the propagation
of matter density waves on the surface of a self-gravitating and differentially
rotating disk. Finally we find a new version of Toomre's stability criterion
for thin disks. We show that EiBI gravity with negative $\chi$ destabilizes all
the rotating thin disks. On the other hand EiBI with positive $\chi$
substantially can suppress the local fragmentation, and has stabilizing effects
against axi-symmetric perturbations. More specifically, we show that only an
annulus remains unstable on the surface of the disk. The width of the annulus
directly depends on the magnitude of $\chi$.
