  As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood
gave asymptotic formulas called the approximate functional equation. In this
paper, we prove an approximate functional equation of the Barnes double
zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty
\sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Also, applying this approximate
functional equation and the van der Corput method, we obtain upper bounds for $
\zeta_2(1/2 + it, \alpha ; v, w) $ and $ \zeta_2(3/2 + it, \alpha ; v, w) $
with respect to $ t $ as $ t \rightarrow \infty $.
