  We consider a non-uniquely ergodic dynamical system given by a
$\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) $\tau$ on a non-empty
compact metrisable space $\Omega$, for some $l\in\N$. 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-2 large
deviation principle, level-1 large deviation principle, convexity properties of
some maps on $\R^n$ for all $n\in\N$. 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 $(f,g)\in \Sigma\times
V\setminus\{0\}$; 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 $V=\textnormal{span}(\{f_n+h_n: n\in\N\})$. 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 measure of maximal entropy.
