Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $μ\in\mathcal{P}(X),\,ν\in\mathcal{P}(Y)$, we propose to study the Monge-Kantorovich problem and its duality from a zero temperature limit of a convex pressure function. We consider the entropy defined by $H(π) = -D_{KL}(π|μ\times ν)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(βA) = \sup_{π\in Π(μ,ν)} \left[ \smallint βA\,dπ+ H(π)\right],\]where $β>0$ and $A=-c$. We will show that it admits a dual formulation and when $β\to+\infty$ we recover the solution for the usual Monge-Kantorovich problem and its Kantorovich duality. Such approach is similar to one which is well known in Thermodynamic Formalism and Ergodic Optimization, where $β$ is interpreted as the inverse of the temperature ($β= \frac{1}{T}$) and $β\to+\infty$ is interpreted as a zero temperature limit.