  Faltings' approach in $p$-adic Hodge theory can be schematically divided into
two main steps: firstly, a local reduction of the computation of the $p$-adic
\'etale cohomology of a smooth variety over a $p$-adic local field to a Galois
cohomology computation and then, the establishment of a link between the latter
and differential forms. These relations are organized through Faltings ringed
topos whose definition relies on the choice of an integral model of the
variety, and whose good properties depend on the (logarithmic) smoothness of
this model. Scholze's generalization for rigid analytic varieties has the
advantage of depending only on the variety (i.e. the generic fibre). Inspired
by Deligne's approach to classical Hodge theory for singular varieties, we
establish a cohomological descent result for the structural sheaf of Faltings
topos, which makes it possible to extend Faltings' approach to any integral
model, i.e. without any smoothness assumption. An essential ingredient of our
proof is a descent result of perfectoid algebras in the arc-topology due to
Bhatt and Scholze. As an application of our cohomological descent, using a
variant of de Jong's alteration theorem for morphisms of schemes, we generalize
Faltings' main $p$-adic comparison theorem to any proper and finitely presented
morphism of coherent schemes over an absolute integral closure of
$\mathbb{Z}_p$ (without any assumption of smoothness) for torsion \'etale
sheaves (not necessarily finite locally constant).
