One of the coarsest topological invariants is the Euler characteristic of cohomology with compact support. For a complex algebraic variety $X$ considered as a set of $\mathbb{C}$-rational points with the classical / Euclidean / analytic topology, for a Zariski closed subset $Z\subseteq X$ with its induced topology, and for the open complement $X\setminus Z$ with its induced topology, there is a long exact sequence of compactly supported cohomology, $$\dots \to H^r_c(X\setminus Z;\mathbb{Z}) \to H^r_c(X;\mathbb{Z}) \to H^r_c(Z;\mathbb{Z}) \xrightarrow{\delta} H^{r+1}_c(X\setminus Z;\mathbb{Z})\to \dots $$ This gives rise to an equality of compactly supported Euler characteristics, $$\chi_c(X) = \chi_c(X\setminus Z) + \chi_c(Z).$$
Thus, for a linearly nondegenerate, irreducible, Zariski closed subset $X$ of $\mathbb{P}^n$ that is smooth, if for every hyperplane section $H$ the open complement $X\setminus H$ has equal Betti numbers, then also the Euler characteristic $\chi_c(H)$ is independent of the choice of hyperplane section.
On the other hand, for a field $k$ of characteristic zero, such as $k=\mathbb{C}$, for an integral closed subscheme $X$ of $\mathbb{P}^n_k$ that is smooth, a general pencil $(H_t)_{t\in \Pi}$ of hyperplane sections $H_t$ will be a Lefschetz pencil. There are many references for this; one reference is Corollary 2.10, p. 46 of the following.
MR2449178 (2009j:32015)
Voisin, Claire
Hodge theory and complex algebraic geometry. II.
Translated from the French by Leila Schneps. Reprint of the 2003 English edition.
Cambridge Studies in Advanced Mathematics, 77.
Cambridge University Press, Cambridge, 2007. x+351 pp.
In positive characteristic, this can fail, although it is often still true. The standard reference is the second volume of SGA 7.
In characteristic $0$, there will be only finitely many elements $t$ of the pencil, say $t\in\{t_1,\dots,t_\delta\}$, such that $H_t$ is singular, and each singular member $H_t$ will have a single ordinary double point. The finite set $\{t_1,\dots,t_\delta\}$ is the discriminant locus. For each such $t$, the Betti numbers of $H_t$ and of the nearby fibers $H_s$ will differ in precisely one degree, coming from a vanishing cycle, so that the difference of Betti numbers is $\pm 1$ (depending on the parity of the cohomological degree of the vanishing cycle). Thus, $\chi_c(H_t)-\chi_c(H_s)$ equals $+1$ or $-1$ for every $t$ in the discriminant locus. The precise cardinality of the discriminant locus is computed in my answer to a previous MathOverflow question: Bounding the number of critical points in a Lefschetz pencil