  We introduce the notion of \textit{relative $L^p$-cohomology} as a
quasi-isometry invariant defined for Gromov-hyperbolic spaces, and apply it to
the problem of quasi-isometry classification of Heintze groups. More precisely,
we explicitly construct non-zero relative $L^p$-cohomology classes on a Heintze
group of the form $\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}$, which gives a way
to prove that the eigenvalues of $\alpha$, up to a scalar multiple, are
invariant by quasi-isometries.
  In the case of degree $1$ we show a relation between the relative and the
classical $L^p$-cohomology.
