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We show that the continuous $L^p$-cohomology of locally compact secondcountable groups is a quasi-isometric invariant. As an application, we provepartial results supporting a positive answer to a question asked by M.~Gromov,suggesting a classical behaviour of continuous $L^p$-cohomology of simple realLie groups. In addition to quasi-isometric invariance, the ingredients are aspectral sequence argument and Pansu's vanishing results for real hyperbolicspaces. In the best adapted cases of simple Lie groups, we obtain nearly halfof the relevant vanishings.

Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings