The Gaussian distribution maximizes entropy for the following functional constraints
$$E(x) = \mu$$ and $$E((x-\mu)^2) = \sigma^2$$
which are just its first and second statistical moments (true parameters, not estimates of them),
as well the constraint that $x$ be included in the support of the probability density, which for the Gaussian is $(-\infty, \infty)$.
Is the above suggesting some sort of link between entropy and moments? By imposing those constraints (knowing the true moments?), we can be assured of maximum entropy as well as our estimated entropy value? Does this suggest that the statistical moments and entropy can be defined by one another and that, if I have the moments, I can calculate the corresponding entropy and vice versa? This would contradict the fact that several distributions with differing moments can have identical entropies though
Source table of distributions, their constraints and supports that provide closed-form analytical maximum entropy solutions. scroll to
Other Examples for the table