Background: The rapid cooling observed in the Cassiopeia A neutron star (Cas A NS) provides one of the most stringent tests for neutron star cooling theory. While the Cooper-pair breaking and formation (PBF) neutrino emission process is a leading candidate, significant theoretical uncertainties remain regarding both the PBF efficiency factor $q$ and the neutron ${}^{3}P_2$ pairing gap.

Purpose: This work aims to explore, in a data-driven manner, how the optimized shape of the neutron ${}^{3}P_2$ pairing gap responds to variations of the phenomenological PBF emissivity parameter $q$ within a fixed and controlled cooling setup.

Methods: We introduce a novel parametrization of the neutron ${}^{3}P_2$ pairing gap, in which each parameter carries direct physical meaning, allowing independent control of the gap amplitude, peak location, width, and asymmetry. Using a newly developed Fortran-based neutron star cooling code and the BSk24 equation of state, we perform systematic parameter-space exploration guided by the Cas A NS observational data. Global optimization is carried out with Optuna's tree-structured Parzen estimator, followed by local refinement using the Nelder-Mead simplex method. Both single-objective optimizations based solely on ${\chi }^2$ and multiobjective optimizations that combine ${\chi }^2$ with a slope-difference metric are examined under identical conditions.

Results: The optimized solutions yield physically reasonable neutron ${}^{3}P_2$ pairing gaps with peak amplitudes ${\mathrm{\Delta }}_{max}\approx 0.5–0.6\mathrm{MeV}$. While the multiobjective formulation explores the parameter space more broadly, the single-objective (${\chi }^2$-only) optimization finally achieves the lowest ${\chi }^2$, reflecting the intrinsically curved nature of neutron star cooling trajectories and the fact that minimizing a slope difference at a single epoch does not necessarily guarantee the best global agreement with the full Cas A NS dataset. Adopting the canonical neutron star mass $M_{\mathrm{NS}}=1.4M_{\odot }$, we find that increasing $q$ drives the optimized gap and critical-temperature profiles toward smoother and more localized shapes and improves consistency with the observed cooling trend. In particular, models with $q\gtrsim 0.4$ reproduce the Cas A NS decline rate within the $1\sigma$ confidence interval, whereas smaller values of $q$ lead to progressively shallower cooling slopes, with the baseline case $q\simeq 0.19$ lying near the $3\sigma$ level within the present setup.

Conclusions: Within this restricted and controlled modeling framework, our results suggest larger effective PBF emissivities than the baseline estimate. However, a statistically robust and less assumption-dependent constraint on $q$ requires a dedicated Bayesian inference that simultaneously accounts for uncertainties in neutron star mass, envelope composition, equation of state, pairing microphysics, and the age offset of Cas A NS. The physically interpretable gap parametrization and hybrid optimization strategy developed here provide a natural foundation for such future Bayesian and machine-learning-based studies of neutron star cooling.

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• Neutron star cooling
• Nuclear astrophysics
• Nuclear forces
• Nuclear matter in neutron stars
• Superfluidity