  We theoretically investigate a $J_1$-$J_3$ classical Heisenberg model on the
breathing pyrochlore lattice, where the nearest-neighbor (NN) exchange
interactions for small and large tetrahedra, $J_1$ and $J_1'$, take different
values due to the breathing bond-alternation and $J_3$ is the third NN
antiferromagnetic interaction along the bond direction. It is found by means of
Monte Carlo simulations that for large $J_3$, a hedgehog lattice, a
three-dimensional periodic array of magnetic monopoles and antimonopoles,
emerges in the form of a quadruple-${\bf Q}$ state characterized by the
ordering vector of ${\bf Q}=(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})$,
being irrespective of the signs of $J_1$ and/or $J_1'$ as long as $J_1\neq
J_1'$. It is also found that in an applied magnetic field, there appear six
quadruple-${\bf Q}$ states depending on the values of $J_1$ and $J_1'$, among
which three phases including the in-field hedgehog-lattice state exhibit
nonzero total chirality $\mbox{\boldmath $\chi$}^{\rm T}$ associated with the
anomalous Hall effect of chirality origin. In the remaining two chiral phases,
which are realized in the presence of ferromagnetic $J_1$ and/or $J_1'$, the
spin structure is not topologically nontrivial, in spite of the fact that
$\mbox{\boldmath $\chi$}^{\rm T} \neq 0$. The role of the topological objects
of the monopoles in $\mbox{\boldmath $\chi$}^{\rm T}$ is also discussed.
