It is a classical result in thermodynamic formalism that, as the temperature goes to zero, the pressure of a potential approaches a slant asymptote. How fast does it get there? For a Walters potential on a mixing subshift of finite type, we show that the rate of convergence, measured on an exponential scale, always exists. Moreover, this rate is strictly positive if and only if the union of the maximal entropy components of the Aubry set is a subshift of finite type. We also establish a dichotomy for Hölder potentials: if this union is a subshift of finite type, the pressure converges to its asymptote exponentially fast, while if it is sofic (but not of finite type), the convergence is polynomial. This is joint work with J.-R. Chazottes, A. Quas, and R. Leplaideur