  Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this
paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave
equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous
degree $\nu\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and
initial data taken from negative order homogeneous Sobolev space $\dot
H^{-\gamma}(\mathbb G), \gamma>0$. In the framework of Sobolev spaces of
negative order, we prove that $p_{\text{Crit}}(Q, \gamma, \nu)
:=1+\frac{2\nu}{Q+2\gamma}$ is the new critical exponent for $\gamma\in (0,
\frac{Q}{2})$. More precisely, we show the global-in-time existence of small
data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, \gamma,
\nu) $ in the energy evolution space $ \mathcal{C}\left([0, T],
H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial
data, we also prove a finite-time blow-up of weak solutions for
$1<p<p_{\text{Crit}}(Q, \gamma, \nu)$. Furthermore, to precisely characterize
the blow-up time, we derive sharp upper bound and lower bound estimates for the
lifespan in the subcritical cases. We emphasize that our results are also new,
even in the setting of higher-order differential operators on $\mathbb{R}^n$,
and more generally, on stratified Lie groups.
