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In this paper we study ergodic optimization problems for subadditivesequences of functions on a topological dynamical system. We prove that for$t\rightarrow \infty$ any accumulation point of a family of equilibrium statesis a maximizing measure. We show that the Lyapunov exponent and entropy ofequilibrium states converge in the limit $t\rightarrow \infty$ to the maximumLyapunov exponent and entropy of maximizing measures.  In the particular case of matrix cocycles we prove that the maximal Lyapunovexponent can be approximated by Lyapunov exponents of periodic trajectoriesunder certain assumptions.

Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent