Understanding the origin of thermostability is of fundamental importance in protein biochemistry. Opposing views on increased or decreased structural rigidity of the folded state have been put forward in this context. They have been related to differences in the temporal resolution of experiments and computations that probe atomic mobility. Here, we find a significant (p = 0.004) and fair (R2 = 0.46) correlation between the structural rigidity of a well-characterized set of 16 mutants of lipase A from Bacillus subtilis (BsLipA) and their thermodynamic thermostability. We apply the rigidity theory-based Constraint Network Analysis (CNA) approach, analyzing directly and in a time-independent manner the statics of the BsLipA mutants. We carefully validate the CNA results on macroscopic and microscopic experimental observables and probe for their sensitivity with respect to input structures. Furthermore, we introduce a robust, local stability measure for predicting thermodynamic thermostability. Our results complement work that showed for pairs of homologous proteins that raising the structural stability is the most common way to obtain a higher thermostability. Furthermore, they demonstrate that related series of mutants with only a small number of mutations can be successfully analyzed by CNA, which suggests that CNA can be applied prospectively in rational protein design aimed at higher thermodynamic thermostability.