© 2022 Elsevier B.V.We explore the possible advantages of extending the standard ΛCDM model by more realistic backgrounds compared to its spatially flat Robertson–Walker (RW) spacetime assumption, while preserving the underpinning physics; in particular, by simultaneously allowing non-zero spatial curvature and anisotropic expansion on top of ΛCDM, viz., the An-oΛCDM model. This is to test whether the latest observational data still support spatial flatness and/or isotropic expansion in this case, and, if not, to explore the roles of spatial curvature and expansion anisotropy (due to its stiff fluid-like behavior) in addressing some of the current cosmological tensions associated with ΛCDM. We first present the theoretical background and explicit mathematical construction of An-oΛCDM; in the simplest manner, combining the simplest anisotropic generalizations of the RW spacetime, viz., the Bianchi type I, V, and IX spacetimes, in one Friedmann equation. Then we constrain the parameters of this model and its particular cases, namely, An-ΛCDM, oΛCDM, and ΛCDM, by using the data sets from different observational probes, viz., Planck cosmic microwave background (CMB) with or without lensing (Lens), baryonic acoustic oscillations (BAO), type Ia Supernovae (SnIa) Pantheon, and cosmic chronometers (CC) data, and discuss the results in detail. Ultimately, we conclude that, within the setup under consideration, (i) the observational data confirm the spatial flatness and isotropic expansion assumptions of ΛCDM, though a very small amount of present-day expansion anisotropy (Ωσ0) cannot be excluded, e.g., Ωσ0≲10−18 (95% C.L.) for An-ΛCDM from CMB+Lens data, (ii) the introduction of spatial curvature or anisotropic expansion, or both, on top ΛCDM does not offer a possible relaxation to the H0 tension, and (iii) the introduction of anisotropic expansion neither affects the closed space prediction from the CMB data nor does it improve the drastically reduced value of H0 led by the closed space. We discuss why it is important and indispensable to maintain the geometric generalization work program, especially in models that offer solutions to cosmological tensions, even though our findings do not appear to favor the geometric generalizations of ΛCDM considered in this work.