  We study the behavior of the pressure function for H\"{o}lder continuous
potentials on mixing subshifts of finite type. The classical theory of
thermodynamic formalism shows that such pressure functions are convex, analytic
and have slant asymptotes. We provide a sharp exponential lower bound on how
fast the pressure function approaches its asymptotes. As a counterpart, we also
show that there is no corresponding upper bound by exhibiting systems for which
the convergence is arbitrarily slow. However, we prove that the exponential
upper bound still holds for a generic H\"{o}lder potential. In addition, we
determine that the pressure function satisfies a coarse uniform convexity
property. Asymptotic bounds and quantitative convexity estimates are the first
additional general properties of the pressure function obtained in the settings
of Bowen and Ruelle since their groundbreaking work more than 40 years ago.
