Generalized quasi-topological gravities: the whole shebang

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