  In this paper, we explore the relativistic quantum motion of spin-zero scalar
particles influenced by rainbow gravity's in the background of a magnetic
solution. Our focus is on the Bonnor-Melvin magnetic space-time, a
four-dimensional solution featuring a non-zero cosmological constant. To
analyze this scenario, we solve the Klein-Gordon equation using two sets of
rainbow functions: (i) $f(\chi)=\frac{1}{(1-\beta\,\chi)}$, $h(\chi)=1$ and
(ii) $f(\chi)=\frac{1}{(1-\beta\,\chi)}=h(\chi)$, where $0 <
\chi(=\frac{|E|}{E_p}) \leq 1$ with $E_p$ being Planck's energy. The resulting
solutions provide the relativistic energy profiles of the scalar particles.
Furthermore, we study the quantum oscillator fields through the Klein-Gordon
oscillator within the same Bonnor-Melvin magnetic space-time background.
Employing $f(\chi)=\frac{1}{(1-\beta\,\chi)}$, $h(\chi)=1$ as the rainbow
function, we obtain the eigenvalue solutions for the quantum oscillator fields.
Notably, we demonstrate that the relativistic energy profile of scalar
particles and the oscillator fields are influenced by the topology of the
geometry and the cosmological constant both are connected with the magnetic
field strength. Additionally, we highlight the impact of the rainbow parameter
on the energy profiles.
