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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-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé 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-Dreyssé model, and the native peptide bonds in the Wako-Saitô-Muñoz-Eaton model.