Summary: This work presents a mesh refinement indicator based on entropy variables, with an application to the compressible Navier-Stokes equations. The entropy variables are shown to satisfy an adjoint equation, an observation that allows recent work in adjoint-based error estimation to be leveraged in constructing a relatively cheap but effective adaptive indicator. The output associated with the entropy-variable adjoint is shown to be the entropy production in the computational domain, including physical viscous dissipation when present, minus entropy transport out of the domain. Adaptation using entropy variables, which is equivalent to adapting on the integrated residual of the entropy transport equation, thus targets areas of the domain responsible for numerical, or spurious, entropy production. Adaptive results for inviscid and viscous aerodynamic examples in two and three dimensions demonstrate performance efficiency on par with output-based adaptation, as measured by errors in various engineering quantities of interest, with the comparative advantage of the proposed approach that no adjoint equations need to be solved.