For a finitely primitive topological Markov shift on a countably infinite number of alphabets
we establish the large deviation principle. More precisely, we assume the existence of
a Gibbs measure for a potential $\phi$ in the sense of Bowen, and prove the level-2
Large Deviation Principles for the distribution of Birkhoff averages under the Gibbs measure,
as well as that of weighted periodic points and iterated pre-images. The rate function is in common,
written with the pressure and the free energy associated with the potential $\phi$.
The Gibbs measure is not assumed to be an equilibrium state for the potential $\phi$,
nor is assumed the existence of an equilibrium state. We provide a sufficient condition
for minimizers of the rate function to be equilibrium states.