The evolution of structural proteins is generally constrained by the folding stability. However, little is known about the particular capacity of viral proteins to accommodate mutations that can potentially affect the protein stability and, in general, the evolution of the protein stability over time. As an illustrative model case, here we investigated the evolution of the stability of the HIV-1 protease (PR), which is a common HIV-1 drug target, under diverse evolutionary scenarios that include (i) intra-host virus evolution in a cohort of 75 patients sampled over time, (ii) intra-host virus evolution sampled before and after specific PR-based treatments and, (iii) inter-host evolution considering extant and ancestral (reconstructed) PR sequences from diverse HIV-1 subtypes. We also investigated the specific influence of currently known HIV-1 PR resistance mutations on the PR folding stability. We found that the HIV-1 PR stability fluctuated over time within a constant and wide range in any studied evolutionary scenario, accommodating multiple mutations that partially affected stability while maintaining activity. We did not identify relationships between change of PR stability and diverse clinical parameters such as viral load, CD4+ T cell counts and a surrogate of time from infection. Counterintuitively, we predicted that near half of the studied HIV-1 PR resistance mutations do not significantly decrease stability, which together with compensatory mutations, would allow the protein to adapt without requiring dramatic stability changes. We conclude that the HIV-1 PR presents a wide structural plasticity to acquire molecular adaptations without affecting the overall evolution of stability.