Let $\mathsf{RO}(X)$ stand for the collection of regular open subsets of a topological space $X$ and let $d(X)$ be its density. It is well-known (see Theorem~3.3 of Hodel's chapter in the Handbook) that every regular space satisfies the inequality $|\mathsf{RO}(X)|\leq 2^{d(X)}$. What is an example of an infinite $T_3$ (regular + $T_1$) space such that $|\mathsf{RO}(X)|<2^{d(X)}$?
I think a relevant observation regarding this question is the following. From Pierce's result on the cardinality of complete Boolean algebras (see here), it can easily be deduced that if $X$ is infinite and $T_2$, then $|\mathsf{RO}(X)|\geq \omega_1$. Thus, $\textsf{CH}$ implies that if $X$ is infinite, $T_3$ and separable, then $|\mathsf{RO}(X)| = 2^{d(X)}$. For this reason, if $X$ satisfies the desired characteristics of the question above, then necessarily $d(X)\geq \omega_1$.