We study the local stability near the maximum figure of merit for the low-dissipation cyclic refrigerator, where the irreversible dissipation occurs not only in the thermal contacts but also the adiabatic strokes. We find that the bounds of the coefficient of performance at maximum figure of merit or maximum cooling rate in presence of internal dissipation are identical to corresponding those in absence of internal dissipation. Using two different scenarios, we prove the existence of a single stable steady state for the refrigerator, and clarify the role of internal dissipation on the stability of thermodynamic steady state, showing that the speed of system evolution to the steady state decreases due to internal dissipation.