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