  We investigate the sharp asymptotic behavior at criticality of the large
fluctuations of extensive observables in renewal models of statistical
mechanics, such as the Poland-Scheraga model of DNA denaturation, the
Fisher-Felderhof model of fluids, the Wako-Sait\^o-Mu\~noz-Eaton model of
protein folding, and the Tokar-Dreyss\'e model of strained epitaxy. These
models amount to Gibbs changes of measure of a classical renewal process and
can be identified with a constrained pinning model of polymers. The extensive
observables that enter the thermodynamic description turn out to be cumulative
rewards corresponding to deterministic rewards that are uniquely determined by
the waiting time and grow no faster than it. The probability decay with the
system size of their fluctuations switches from exponential to subexponential
at criticality, which is a regime corresponding to a discontinuous
pinning-depinning phase transition. We describe such decay by proposing a
precise large deviation principle under the assumption that the subexponential
correction term to the waiting time distribution is regularly varying. This
principle is in particular used to characterize the fluctuations of the number
of renewals, which measures the DNA-bound monomers in the Poland-Scheraga
model, the particles in the Fisher-Felderhof model and the Tokar-Dreyss\'e
model, and the native peptide bonds in the Wako-Sait\^o-Mu\~noz-Eaton model.
