  We study the Bishop-Phelps-Bollob\'as property for operators between Banach
spaces. Sufficient conditions are given for generalized direct sums of Banach
spaces with respect to a~uniformly monotone Banach sequence lattice to have the
approximate hyperplane series property. This result implies that
Bishop-Phelps-Bollob\'as theorem holds for operators from $\ell_1$ into such
direct sums of Banach spaces. We also show that the direct sum of two spaces
with the approximate hyperplane series property has such property whenever the
norm of the direct sum is absolute.
