Let $$ f(y | \eta) = h(y) \exp\left( \eta^\top T(y) + A(\eta) \right)$$ be the exponential family with base density/pmf $h$, sufficient statistic $T$, log partition function $A$ and natural parameter $\eta$. If it helps, we may restrict it to the case where $\eta$ is scalar. Obviously $h$ must be nonnegative. One can show that $A$ must also be convex.
The Wikipedia page states:
The function $A(\eta)$ ... is automatically determined once the other functions have been chosen, since it must assume a form that causes the distribution to be normalized (sum or integrate to one over the entire domain).
Bold s on functions by me for emphasis.
My Question:
Given some fixed $T$, am I free to choose any convex $A$ such that there exists an $h$ such that the resulting function is a valid pdf/pmf? More precisely, given some $T$ consider the set $$\mathcal{A}_T = \{ A:\mathbb{R} \to \mathbb{R} \mid \exists \, h:\mathbb{R} \to \mathbb{R} \mid A(\eta) = \log\int h(y)\exp \left(\eta T(y) \right)\, dy\}.$$ Is there a useful alternative characterisation of this set?