  Higher-dimensional topological phases play a key role in understanding the
lower-dimensional topological phases and the related topological responses
through a dimensional reduction procedure. In this work, we present a
Dirac-type model of four-dimensional (4D) $\mathbb{Z}_2$ topological insulator
(TI) protected by $\mathcal{CP}$-symmetry, whose 3D boundary supports an odd
number of Dirac cones. A specific perturbation splits each bulk massive Dirac
cone into two valleys separated in energy-momentum space with opposite second
Chern numbers, in which the 3D boundary modes become a nodal sphere or a Weyl
semimetallic phase. By introducing the electromagnetic (EM) and pseudo-EM
fields, exotic topological responses of our 4D system are revealed, which are
found to be described by the (4+1)D mixed Chern-Simons theories in the
low-energy regime. Notably, several topological phase transitions occur from a
$\mathcal{CP}$-broken $\mathbb{Z}_2$ TI to a $\mathbb{Z}$ TI when the bulk gap
closes by giving rise to exotic double-nodal-line/nodal-hyper-torus gapless
phases. Finally, we propose to probe experimentally these topological effects
in cold atoms.
