  In this paper, we study the stability of the conical K\"ahler-Ricci flows on
Fano manifolds. That is, if there exists a conical K\"ahler-Einstein metric
with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$
sufficiently close to $\beta$, the corresponding conical K\"ahler-Ricci flow
converges to a conical K\"ahler-Einstein metric with cone angle $2\pi\beta'$
along the divisor. Here, we only use the condition that the Log Mabuchi energy
is bounded from below. This is a weaker condition than the properness that we
have adopted to study the convergence before. As corollaries, we give parabolic
proofs of Donaldson's openness theorem and his existence conjecture for the
conical K\"ahler-Einstein metrics with positive Ricci curvatures.
