pr.probability - for a natural exponential family, A is the cumulant function of h?

Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) exp(langle theta, x rangle - A(theta))$
defines a family of distributions for $X$, parametrized by $theta$, then $A$ is the cumulant generating function of $h(X)$. It seems like this should be easy to prove if it's true, but I don't see how to proceed. Any ideas/references?