Let $G$ be a topological group. It has a classifying space $BG$, which has homology groups $H_{*}BG$. Changing the topology of $G$ affects the space $BG$ and hence its homology groups.

For example the group $\mathbb{R}$ with its usual topology has $H_{*}B\mathbb{R}\simeq H_{*}pt$. Changing the topology to be much finer, namely the discrete topology, results in a topological group $\mathbb{R}^{\delta}$ that has nontrivial $H_{1}$.

Question: Is it possible to go the other way? I.e. does there exist a topological group $G$, which has a second topology, coarser than the first, that makes the group homology larger?

There are many ways to make "larger" precise--I'm most interested in the case where initial topological group is acyclic. Then the meaning of "larger" is clear.

Motivation: It would help answer this question.