  The solutions of many issues, of the ongoing efforts to make deformed
graphene a tabletop quantum field theory in curved spacetimes, are presented. A
detailed explanation of the special features of curved spacetimes, originating
from embedding portions of the Lobachevsky plane into $\mathbf{R}^3$, is given,
and the special role of coordinates for the physical realizations in graphene,
is explicitly shown, in general, and for various examples. The Rindler
spacetime is reobtained, with new important differences with respect to earlier
results. The de Sitter spacetime naturally emerges, for the first time, paving
the way to future applications in cosmology. The role of the BTZ black hole is
also briefly addressed. The singular boundary of the pseudospheres, "Hilbert
horizon", is seen to be closely related to event horizon of the Rindler, de
Sitter, and BTZ kind. This gives new, and stronger, arguments for the Hawking
phenomenon to take place. An important geometric parameter, $c$, overlooked in
earlier work, takes here its place for physical applications, and it is shown
to be related to graphene's lattice spacing, $\ell$. It is shown that all
surfaces of constant negative curvature, ${\cal K} = -r^{-2}$, are unified, in
the limit $c/r \to 0$, where they are locally applicable to the Beltrami
pseudosphere. This, and $c = \ell$, allow us a) to have a phenomenological
control on the reaching of the horizon; b) to use spacetimes different than
Rindler for the Hawking phenomenon; c) to approach the generic surface of the
family. An improved expression for the thermal LDOS is obtained. A non-thermal
term for the total LDOS is found. It takes into account: a) the peculiarities
of the graphene-based Rindler spacetime; b) the finiteness of a laboratory
surface; c) the optimal use of the Minkowski quantum vacuum, through the choice
of this Minkowski-static boundary.
