Gradient and Hessian-Based temperature estimator in lattice gauge theories: a diagnostic tool for stability and consistency in numerical simulations

We present a field configuration-based temperature estimator in lattice gauge theories, constructed from the gradient and Hessian of the Euclidean action. Adapted from geometric formulations of entropy in classical statistical mechanics, this estimator provides a gauge-invariant, non-kinetic diagnostic of thermodynamic consistency in Monte Carlo simulations. We validate the method in compact U(1) lattice gauge theories across one, two, and four dimensions, comparing the estimated configurational temperature with the conventional temperature set by the temporal extent of the lattice. Our results show that the estimator accurately reproduces the input temperature and remains robust across a range of lattice volumes and coupling strengths. The temperature estimator offers a general-purpose diagnostic for lattice field theory simulations, with potential applications to non-Abelian theories, anisotropic lattices, and real-time monitoring in hybrid Monte Carlo algorithms.