  Fracton models exhibit a variety of exotic properties and lie beyond the
conventional framework of gapped topological order. In a previous work, we
generalized the notion of gapped phase to one of foliated fracton phase by
allowing the addition of layers of gapped two-dimensional resources in the
adiabatic evolution between gapped three-dimensional models. Moreover, we
showed that the X-cube model is a fixed point of one such phase. In this paper,
according to this definition, we look for universal properties of such phases
which remain invariant throughout the entire phase. We propose multi-partite
entanglement quantities, generalizing the proposal of topological entanglement
entropy designed for conventional topological phases. We present arguments for
the universality of these quantities and show that they attain non-zero
constant value in non-trivial foliated fracton phases.
