  We compare the marked length spectra of some pairs of proper and cocompact
cubical actions of a non-virtually cyclic group on $\text{CAT}(0)$ cube
complexes. The cubulations are required to be virtually co-special, have the
same sets of convex-cocompact subgroups, and admit a contracting element. There
are many groups for which these conditions are always fulfilled for any pair of
cubulations, including non-elementary cubulable hyperbolic groups, many
cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter
groups.
  For these pairs of cubulations, we study the Manhattan curve associated to
their combinatorial metrics. We prove that this curve is analytic and convex,
and a straight line if and only if the marked length spectra are homothetic.
The same result holds if we consider invariant combinatorial metrics in which
the lengths of the edges are not necessarily one. In addition, for their
standard combinatorial metrics, we prove a large deviation theorem with
shrinking intervals for their marked length spectra. We deduce the same result
for pairs of word metrics on hyperbolic groups.
  The main tool is the construction of a finite-state automaton that
simultaneously encodes the marked length spectra of both cubulations in a
coherent way, in analogy with results about (bi)combable functions on
hyperbolic groups by Calegari and Fujiwara. The existence of this automaton
allows us to apply the machinery of thermodynamic formalism for suspension
flows over subshifts of finite type, from which we deduce our results.
