  In this paper we study the thermodynamic formalism of strongly transitive
endomorphisms $f$, focusing on the set all expanding measures. In case $f$ is a
non-flat $C^{1+}$ map defined on a Riemannian manifold, these are invariant
probability measures with all its Lyapunov exponents positive. Given a H\"older
continuous potential $\varphi$ we prove the uniqueness of the equilibrium state
among the space of expanding measures. Moreover, we show that the existence of
an expanding measure $\mu$ maximizing the entropy on the the space of expanding
measures implies the existence and uniqueness of equilibrium state
$\mu_{\varphi}$ on the space of expanding measures for any H\"older continuous
potential $\varphi$ with a small oscillation $\text{osc
}\varphi=\sup\varphi-\inf\varphi$. As some applications, we prove that
Collet-Eckmann quadratic maps does not admit phase transition for H\"older
potential, and show that for Viana maps and every H\"older continuous potential
of sufficiently small oscillation has a unique equilibrium state.
