Real critical points of $T$-polynomials that are sums of squared monomials and topology of $T$-hypersurfaces

We study the topology of the real algebraic hypersurfaces in ${\mathbb{P}}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on various sums of their Betti numbers. We then discuss the sharpness of those bounds.