We here show how evaporation/condensation processes lead to efficient heat spreading along a liquid/gas interface, thereby damping thermal fluctuations and hindering thermocapillary flows. This mechanism acts as an effective thermal conductivity of the gas phase, which is shown to diverge when the latter is made of pure vapor. Our simple (fitting-parameter–free) theory nicely agrees with measurements of critical conditions for Bénard-Marangoni instability in drying liquid films. Heat spreading is also shown to strongly affect wavelength selection in the nonlinear regime. In addition to providing a quantitative framework for analyzing transitions between complex evaporation-driven patterns, this also opens new perspectives for better controlling deposition techniques based on drying.