Can $\mathrm{HOD}^{L(\mathbb R)}$ change between models with large cardinals? It is a celebrated result that if there is a proper class of Woodin cardinals, then the theory $L(\mathbb R)$ cannot be changed by set-sized forcing. Assume there is a proper class of Woodins. Let $\mathbb P \in V$ and let $G \subseteq \mathbb P$ be generic. Is possible that $\mathrm{HOD}^{L(\mathbb R^V)} \not= \mathrm{HOD}^{L(\mathbb R^{V[G]})}$?

Woodin showed that under $\mathrm{AD}^{L(\mathbb R)}$, $\Theta$ is the least Woodin cardinal in $\mathrm{HOD}^{L(\mathbb R)}$. See Koellner-Woodin in the Handbook of Set Theory, and Theorem 8.23 in Steel’s chapter of the same book.

Thus if we force to change the value of $\Theta^{L(\mathbb R)}$, by collapsing it to countable, then the new $\mathrm{HOD}^{L(\mathbb R)}$ has a different least Woodin cardinal.