Scaling relations between anomalous signatures in low-dimensional thermal transport, i.e., length-dependence and non-Brownian growth, call for a universal interpretation. An entropic Burgers-Kardar-Parisi-Zhang class is derived based on fluctuations of stochastic entropy production in heat conduction. A generic scaling framework of the anomalous thermal conductivity is thereafter established from the dynamical scaling theory of fluctuating hydrodynamics. This scaling framework demonstrates that two sound scaling laws between the length-dependent thermal conductivity and mean square of displacement are traceable to Galilean invariance. It also expects other anomalous behaviors including the deviation between equilibrium and non-equilibrium schemes, dimensionality-dependence, logarithmic divergence, failure of the “transit time” truncation, and non-Kardar-Parisi-Zhang scaling. Our results and their coincidences with existing investigations illustrate that Langevin-like formalisms for stochastic thermodynamics can be applied to energy transport in low-dimensional and mesoscopic systems.