Power-law distributions are widely observed in complex systems, yet establishing their thermodynamic consistency remains a theoretical challenge. In this paper, we present a thermodynamic framework for power-law statistics based on the \textit{renormalized entropy} $s_{2-q}$. Derived from the asymptotic scaling of the combinatorial $q$-factorial, this quantity yields a stable thermodynamic limit, remaining finite ($O(N^0)$) for systems with strong correlations. Furthermore, we clarify the physical origin of the nonlinearity parameter $q$ through the concept of \textit{Varentropy} (Variance of Entropy). By unifying the macroscopic variational principle with the microscopic Superstatistics framework, we derive the relation $|q-1| \simeq 1/C$, where $C$ is the heat capacity of the reservoir. This result suggests that power-law statistics provides a thermodynamic description of finite systems, where the finite heat capacity of the heat bath necessitates a generalization beyond the standard Boltzmann-Gibbs limit ($C \to \infty$).

翻译：幂律分布广泛存在于复杂系统中，但建立其热力学一致性仍是一个理论难题。本文基于重整化熵 $s_{2-q}$，提出了一套适用于幂律统计的热力学框架。该量由组合 $q$-阶乘的渐近标度推导得出，能够产生稳定的热力学极限，并在强关联系统中保持有限性（$O(N^0)$）。此外，我们通过变熵（熵的方差）概念阐明了非线性参数 $q$ 的物理起源。通过统一宏观变分原理与微观超统计框架，我们推导出关系式 $|q-1| \simeq 1/C$，其中 $C$ 为热库的热容。这一结果表明，幂律统计为有限系统提供了一种热力学描述，其中热浴的有限热容要求超越标准玻尔兹曼-吉布斯极限（$C \to \infty$）进行推广。