In recent years, lipid cubic nanoparticles have emerged as promising nanocarriers for drug delivery, due to the several advantages they exhibit with respect to other lipid systems. Here, we report on lipid cubic nanoparticles stabilized by PNIPAM-based amphiphilic block copolymers, specifically, poly(N, N-dimethylacrylamide)-block-poly(N-isopropylacrylamide) (PDMA-b-PNIPAM), as a new class of drug delivery systems (DDS). In vitro studies on the internalization efficiency of the DDS towards two types of human cancer cells (colon HCT-116 and bladder T24 cells), carried out employing a set of sensitive techniques (confocal laser scanning microscopy (CLSM), flow cytometry, scanning electron microscopy (SEM), fluorescence spectroscopy), highlight a prominent role of PDMA-b-PNIPAM stabilizer in enhancing the uptake of cubosomes, compared to the standard Pluronic F127-based formulations. The drug delivery potential of cubosomes, tested by encapsulating a chemotherapeutic drug, camptothecin (CPT), and conducting cytotoxicity studies against 2D plated cells and 3D spheroids, confirm that PDMA-b-PNIPAM-stabilized cubosomes improve the efficacy of treatment with CPT. The origin of this effect lies in the higher lipophilicity of the stabilizer, as we confirm by studying the interaction between the cubosomes and biomimetic membranes of lipid vesicles with Small Angle X-Ray Scattering (SAXS) and CLSM experiments. These results corroborate our fundamental understanding of the interaction between cubosomes and cells, and on the role of polymer to formulate lipid cubic nanoparticles as DDS.