  In 2009 Ho\v{r}ava proposed a power-counting renormalizable quantum gravity
theory. Afterwards a term in the action that softly violates the detailed
balance condition has been considered with the attempt of obtaining a more
realistic theory in its IR-limit. This term is proportional to $\omega
R^{(3)}$, where $\omega$ is a constant parameter and $R^{(3)}$ is the spatial
Ricci scalar. In this paper we derive constraints on this IR-modified
Ho\v{r}ava theory using the late-time cosmic accelerating expansion
observations. We obtain a lower bound of $|\omega|$ that is nontrivial and
depends on $\Lambda_W$, the cosmological constant of the three dimensional
spatial action in the Ho\v{r}ava gravity. We find that to preserve the detailed
balance condition, one needs to fine-tune $\Lambda_W$ such that $- 2.29\times
10^{-4}< (c^2 \Lambda_W)/(H^2_0 \currentDE) - 2 < 0 $, where $H_0$ and
$\currentDE$ are the Hubble parameter and dark energy density fraction in the
present epoch, respectively. On the other hand, if we do not insist on the
detailed balance condition, then the valid region for $\Lambda_W$ is much
relaxed to $-0.39< (c^2 \Lambda_W)/(H^2_0 \currentDE) - 2 < 0.12$. We find that
although the detailed balance condition cannot be ruled out, it is strongly
disfavored.
