  In our investigation, we explore the quantum dynamics of charge-free scalar
particles through the Klein-Gordon equation within the framework of rainbow
gravity's, considering the Bonnor-Melvin-Lambda (BML) space-time background.
The BML solution is characterized by the magnetic field strength along the axis
of symmetry direction which is related with the cosmological constant $\Lambda$
and the topological parameter $\alpha$ of the geometry. The behavior of
charge-free scalar particles described by the Klein-Gordon equation is
investigated, utilizing two sets of rainbow functions: (i)
$f(\chi)=\frac{(e^{\beta\,\chi}-1)}{\beta\,\chi}$,\, $h(\chi)=1$ and (ii)
$f(\chi)=1$,\, $h(\chi)=1+\frac{\beta\,\chi}{2}$. Here $0 <
\Big(\chi=\frac{|E|}{E_p}\Big) \leq 1$ with $E$ represents the particle's
energy, $E_p$ is the Planck's energy, and $\beta$ is the rainbow parameter. We
obtain the approximate analytical solutions for the scalar particles and
conduct a thorough analysis of the obtained results. Afterwards, we study the
quantum dynamics of quantum oscillator fields within this BML space-time,
employing the Klein-Gordon oscillator. Here also, we choose the same sets of
rainbow functions and obtained approximate eigenvalue solution for the
oscillator fields. Notably, we demonstrate that the relativistic approximate
energy profiles of charge-free scalar particles and oscillator fields get
influenced by the topology of the geometry and the cosmological constant.
Furthermore, we show that the energy profiles of scalar particles get
modifications by the rainbow parameter and the quantum oscillator fields by
both the rainbow parameter and the frequency of oscillation
