  Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian
varieties of dimension $g$. The arithmetic height, or arithmetic volume, of
$\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the
metrized Hodge bundle $\overline{\omega}_g$ on $\overline{\mathcal{A}}_g$. In
1999, K\"uhn proved a formula for the arithmetic volume of
$\overline{\mathcal{A}}_1$ in terms of special values of the Riemann zeta
function. In this article, we generalize his result to the case $g=2$.
