  Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^{3}$. We show
the occurrence of log-logarithmic fluctuations around the leading order for the
volume of intrinsic balls in $\mathcal{U}$. As an application, we obtain
similar fluctuations for the quenched heat kernel of the simple random walk on
$\mathcal{U}$.
