  Generalized quasi-topological gravities (GQTGs) are higher-curvature
extensions of Einstein gravity in $D$-dimensions. Their defining properties
include possessing second-order linearized equations of motion around maximally
symmetric backgrounds as well as non-hairy generalizations of Schwarzschild's
black hole characterized by a single function, $f(r)\equiv -
g_{tt}=g_{rr}^{-1}$, which satisfies a second-order differential equation. In
arXiv:1909.07983 GQTGs were shown to exist at all orders in curvature and for
general $D$. In this paper we prove that, in fact, $n-1$ inequivalent classes
of order-$n$ GQTGs exist for $D\geq 5$. Amongst these, we show that one -- and
only one -- type of densities is of the Quasi-topological kind, namely, such
that the equation for $f(r)$ is algebraic. Our arguments do not work for $D=4$,
in which case there seems to be a single unique GQT density at each order which
is not of the Quasi-topological kind. We compute the thermodynamic charges of
the most general $D$-dimensional order-$n$ GQTG, verify that they satisfy the
first law and provide evidence that they can be entirely written in terms of
the embedding function which determines the maximally symmetric vacua of the
theory.
