  We study the geodesic deviation (GD) equation in a generalized version of the
S\'{a}ez--Ballester (SB) theory in arbitrary dimensions. We first establish a
general formalism and then restrict to particular cases, where (i) the
matter-energy distribution is that of a perfect fluid, and (ii) the spacetime
geometry is described by a vanishing Weyl tensor. Furthermore, we consider the
spatially flat FLRW universe as the background geometry.
  Based on this setup, we compute the GD equation as well as the convergence
condition associated with fundamental observers and past directed null vector
fields.
  Moreover, we extend that framework and extract the corresponding geodesic
deviation in the \emph{modified} S\'{a}ez--Ballester theory (MSBT), where the
energy-momentum tensor and potential emerge strictly from the geometry of the
extra dimensions. In order to examine our herein GD equations, we consider two
novel cosmological models within the SB framework. Moreover, we discuss a few
quintessential models and a suitable phantom dark energy scenario within the
mentioned SB and MSBT frameworks.
  Noticing that our herein cosmological models can suitably include the present
time of our Universe, we solve the GD equations analytically and/or
numerically. By employing the correct energy conditions plus recent
observational data, we consistently depict the behavior of the deviation vector
$\eta(z)$ and the observer area distance $r_0(z)$ for our models. Concerning
the Hubble constant problem, we specifically focus on the observational data
reported by the Planck collaboration and the SH0ES collaboration to depict
$\eta(z)$ and $r_0(z)$ for our herein phantom model.
  Subsequently, we contrast our results with those associated with the
$\Lambda$CDM model.
