<p>Consistency constraints for low-energy theories, especially those lacking Lorentz invariance, have recently garnered attention. Building on results from black hole thermodynamics, we investigate the conjecture that leading symmetry-preserving irrelevant deformations of a conformal field theory (CFT) in the infrared must increase the system’s entropy. We show that this entropy-positivity conjecture is equivalent to a decrease in the thermal grand potential at a fixed temperature. We then evaluate this proposal against various known positivity bounds and other physical constraints on effective theories: for <i>U</i>(1) Goldstone bosons with a quartic self-interaction at (non-)zero chemical potential, for the Euler-Heisenberg model, for the <i>O</i>(<i>N</i>) nonlinear sigma model in (2 + 1)<i>D</i>, and for <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi>T</mi> <mover accent="true"> <mi>T</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( T\overline{T} \)</EquationSource> </InlineEquation> deformations of the 2D Ising CFT. We find broad agreement with the entropy-positivity conjecture and we discuss test cases where the conjecture is not expected to apply, such as deformations that break internal symmetries of the CFT.</p>