Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $Δ_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $Δ_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $Δ_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $Δ_{g, w}$ in terms of the combinatorics of $w$.