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We explore the possible advantages of extending the $\Lambda$CDM model bymore realistic backgrounds compared to its spatially flat RW spacetimeassumption, while preserving the underpinning physics; in particular, bysimultaneously allowing non-zero spatial curvature and anisotropic expansion ontop of it, viz., the An-$o\Lambda$CDM model. This is to test whether the latestdata support spatial flatness and/or isotropic expansion, and, if not, toexplore the roles of spatial curvature and expansion anisotropy (due to itsstiff fluid-like behavior) in addressing some of the cosmological tensions. Wefirst present the theoretical background and explicit mathematical constructionof An-$o\Lambda$CDM; combining the simplest anisotropic generalizations of theRW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrainthis model and its particular cases, viz., An-$\Lambda$CDM, $o\Lambda$CDM, and$\Lambda$CDM, by using the data sets from different probes, viz., PlanckCMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results.Ultimately, we conclude that, within the setup under consideration, (i) thedata confirm the spatial flatness and isotropic expansion, though a very smallamount of present-day expansion anisotropy cannot be excluded, e.g.,$\Omega_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$\Lambda$CDM fromCMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion,or both, on top $\Lambda$CDM does not offer a possible relaxation to the $H_0$tension, and (iii) the introduction of anisotropic expansion neither affectsthe closed space prediction from CMB(+Lens) nor does it improve the drasticallyreduced value of $H_0$ led by the closed space. We discuss why it is importantand indispensable to maintain the geometric generalization work program,especially in models that offer solutions to cosmological tensions. [abridged]

Testing spatial curvature and anisotropic expansion on top of the $\Lambda$CDM model