  Khovanov introduced a cohomology theory for oriented classical links whose
graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is
functorial for link cobordisms between classical links, we obtain an invariant
of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering
the surface-knot as a link cobordism between empty links. In this paper, we
define an invariant of a surface-knot which is a generalization of the
Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any
$T^2$-knot has the trivial Khovanov-Jacobsson number.
