A model of proteome-associated chemical energetic costs of cells is derived from protein-turnover kinetics and protein folding. Minimization of the proteostatic maintenance cost can explain a range of trends of proteomes and combines both protein function, stability, size, proteostatic cost, temperature, resource availability, and turnover rates in one simple framework. We then explore the ansatz that the chemical energy remaining after proteostatic maintenance is available for reproduction (or cell division) and thus, proportional to organism fitness. Selection for lower proteostatic costs is then shown to be significant vs. typical effective population sizes of yeast. The model explains and quantifies evolutionary conservation of highly abundant proteins as arising both from functional mutations and from changes in other properties such as stability, cost, or turnover rates. We show that typical hypomorphic mutations can be selected against due to increased cost of compensatory protein expression (both in the mutated gene and in related genes, i.e. epistasis) rather than compromised function itself, although this compensation depends on the protein's importance. Such mutations exhibit larger selective disadvantage in abundant, large, synthetically costly, and/or short-lived proteins. Selection against increased turnover costs of less stable proteins rather than misfolding toxicity per se can explain equilibrium protein stability distributions, in agreement with recent findings in E. coli. The proteostatic selection pressure is stronger at low metabolic rates (i.e. scarce environments) and in hot habitats, explaining proteome adaptations towards rough environments as a question of energy. The model may also explain several trade-offs observed in protein evolution and suggests how protein properties can coevolve to maintain low proteostatic cost.