@YCor : Thank you for the answer, although I don't understand the details. Which rigidity results are you thinking of ? Could you be a bit more specific ?

Dear Ian, I checked in detail this morning your argument. There is one thing which is not clear to me(in the last part). Two metric on $M$ are said to be $C$-close if there is a diffeomorphism $f$ of $M$ such that both $f$ and $f^{-1}$ are $e^C$-Lipschitz, but to get the bound on $L_g(\gamma)$ one would would have to be sure that $f$ preserves the class of $\gamma$. We might need here something like the finiteness of the mapping class group of $M$ to conclude right ?

Thank you Ian, Belgradek's paper answer the very first question I had in mind, namely how different can two negatively curved metric be on the same underlying manifold.