Is it true that every pseudo-Anosov diffeomorphism is not topologically stable?

Question

I have received a report on my recent paper and the referee has written that every pseudo-Anosov on a compact smooth manifold is not topologically stable. I don't know the proof. Would you please help me to understand better. Also it is not clear for me definition of pseudo-Anosov and its relation with Anosov diffeomorphism, but $f:X\to X$ is called topologically stable if for every $\epsilon>0$, there is $\delta>0$ such that if $d(f, g)<\delta$, then there is a continuous map $h:X\to X$ such that $d(h, id_X)<\epsilon$ and $f\circ h= h\circ g$. Thanks a lot.