Let $A$ be an almost commutative algebra and write $A_0 = \text{gr} \, A/ \oplus_{i > 0} \text{gr}_i \, A$. At the bottom of p. 16 here, the author says the following:
... recall first that the scheme $\text{Spec} \, $(gr $A$) has a natural $\mathbb{G}_m$-action, and is cone-scheme over $\text{Spec} \,(A_0)$. Consider Zariski cone-topology on $\text{Spec} \, $(gr $A$), i.e. the topology generated by open cone-subsets $U \subseteq \text{Spec} \, (\text{gr} \, A)$.
I'd be much obliged if someone could explain what's meant by the terms "cone-scheme" and "cone-subset". The former is defined in this paper, but not very clearly, and the latter is not defined at all. (My guess is that it means affine opens $U$ with the property that $U$ is a cone-scheme over $\text{Spec} \, A_0$.)
"Consider the Zariski cone topology on $X$" means consider the topology generated by sets of the form $U\subset X$ where $U$ is Zariski-open and $\Bbb G_m$-equivariant. This is a coarser topology than the usual Zariski topology (ie fewer open sets).
A "cone-scheme" is a scheme $X$ that comes with a specified action of $\Bbb G_m$. Saying that $X$ is a cone-scheme over $S$ means that there's map of schemes $X\to S$ such that $S$ is the fixed point locus of the $\Bbb G_m$ action, and the fibers of $X\to S$ are $\Bbb G_m$ equivariant.