Protein mutational landscapes are sculpted by the impacts of the resulting amino acid substitutions on the protein's stability and folding or aggregation kinetics. These properties can, in turn, be modulated by the composition and activities of the cellular proteostasis network. Heat shock factor 1 (HSF1) is the master regulator of the cytosolic and nuclear proteostasis networks, dynamically tuning the expression of cytosolic and nuclear chaperones and quality control factors to meet demand. Chronic increases in HSF1 levels and activity are prominent hallmarks of cancer cells. One plausible explanation for this observation is that the consequent upregulation of proteostasis factors could biophysically facilitate the acquisition of oncogenic mutations. Here, we experimentally evaluate the impacts of chronic HSF1 activation on the mutational landscape accessible to the quintessential oncoprotein p53. Specifically, we apply quantitative deep mutational scanning of p53 to assess how HSF1 activation shapes the mutational pathways by which p53 can escape cytotoxic pressure conferred by the small molecule nutlin-3, which is a potent antagonist of the p53 negative regulator MDM2. We find that activation of HSF1 broadly increases the fitness of dominant-negative substitutions within p53. This effect of HSF1 activation was particularly notable for non-conservative, biophysically unfavorable amino acid substitutions within buried regions of the p53 DNA-binding domain. These results indicate that chronic HSF1 activation profoundly shapes the oncogenic mutational landscape, preferentially supporting the acquisition of cancer-associated substitutions that are biophysically destabilizing. Along with providing the first experimental and quantitative insights into how HSF1 influences oncoprotein mutational spectra, these findings also implicate HSF1 inhibition as a strategy to reduce the accessibility of mutations that drive chemotherapeutic resistance and metastasis.