  The interplay of quantum and classical fluctuations in the vicinity of a
quantum critical point (QCP) gives rise to various regimes or phases with
distinct quantum character. In this work, we show that the R\'enyi entropy is a
precious tool to characterize the phase diagram of critical systems not only
around the QCP but also away from it, thanks to its capability to detect the
emergence of local order at finite temperature. For an efficient evaluation of
the R\'enyi entropy, we introduce a new algorithm based on a path integral
Langevin dynamics combined with a previously proposed thermodynamic integration
method built on regularized paths. We apply this framework to study the
critical behavior of a linear chain of anharmonic oscillators, a particular
realization of the $\phi^4$ model. We fully resolved its phase diagram, as a
function of both temperature and interaction strength. At finite temperature,
we find a sequence of three regimes - para, disordered and quasi long-range
ordered -, met as the interaction is increased. The R\'enyi entropy divergence
coincides with the crossover between the para and disordered regime, which
shows no temperature dependence. The occurrence of quasi long-range order, on
the other hand, is temperature dependent. The two crossover lines merge in
proximity of the QCP, at zero temperature, where the R\'enyi entropy is sharply
peaked. Via its subsystem-size scaling, we confirm that the transition belongs
to the two-dimensional Ising universality class. This phenomenology is expected
to happen in all $\phi^4$-like systems, as well as in the elusive water ice
transition across phases VII, VIII and X.
