  Optimization problems have been the subject of statistical physics
approximations. A specially relevant and general scenario is provided by
optimization methods considering tradeoffs between cost and efficiency, where
optimal solutions involve a compromise between both. The theory of Pareto (or
multi objective) optimization provides a general framework to explore these
problems and find the space of possible solutions compatible with the
underlying tradeoffs, known as the {\em Pareto front}. Conflicts between
constraints can lead to complex landscapes of Pareto optimal solutions with
interesting implications in economy, engineering, or evolutionary biology.
Despite their disparate nature, here we show how the structure of the Pareto
front uncovers profound universal features that can be understood in the
context of thermodynamics. In particular, our study reveals that different
fronts are connected to different classes of phase transitions, which we can
define robustly, along with critical points and thermodynamic potentials. These
equivalences are illustrated with classic thermodynamic examples.
