We consider a non-uniquely ergodic dynamical system given by a Z(l)-action (or Nu boolean OR {0}(l) -action) tau on a non-empty compact metrisable space Omega, for some l epsilon N. Let (D) denote the following property: the graph of the restriction of the entropy map h(tau) to the set of ergodic states is dense in the graph of h(tau). We assume that h(tau) is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map P-tau on some sets dense in the space C(Omega) of real-valued continuous functions on Omega, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on R-n for all n epsilon N. The one involving the Gateaux differentiability of P-tau is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: we show that for each non-empty sigma-compact subset Sigma of C(Omega), D is equivalent to the existence of an infinite dimensional vector space V dense in C(Omega) such that f + g has a unique equilibrium state for all (f, g) epsilon Sigma x V\{0}; any Schauder basis. (f(n)) of C(Omega) whose linear span contains Sigma admits an arbitrary small perturbation. (h(n)) so that one can take V = span ({f(n) + h(n) : n epsilon N }). Taking Sigma = {0}, the existence of an infinite dimensional vector space dense in C(Omega) constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.