  We consider Reissner-Nordstr\"{o}m black holes surrounded by quintessence
where both a non-extremal event horizon and a cosmological horizon exist
besides an inner horizon ($-1\leq \omega <-1/3$). We determine new extreme
black hole solutions that generalize the Nariai horizon to asymptotically de
Sitter-like solutions for any order relation between the squares of the charge
$q^2$ and the mass parameter $M^2$ provided $q^2$ remains smaller than some
limit, which is larger than $M^2$. In the limit case $q^2=9\omega^2
M^2/(9\omega^2-1)$, we derive the general expression of the extreme
cosmo-black-hole, where the three horizons merge, and discuss some of its
properties. We also show that the endpoint of the evaporation process is
independent of any order relation between $q^2$ and $M^2$. The Teitelboim's
energy and Padmanabhan's energy are related by a nonlinear expression and are
shown to correspond to different ensembles. We also determine the enthalpy $H$
of the event horizon, as well as the effective thermodynamic volume which is
the conjugate variable of the negative quintessential pressure, and show that
in general the mass parameter and the Teitelboim's energy are different from
the enthalpy and internal energy; only in the cosmological case, that is, for
Reissner-Nordstr\"{o}m-de Sitter black hole we have $H=M$. Generalized Smarr
formulas are also derived. It is concluded that the internal energy has a
universal expression for all static charged black holes, with possibly a
variable mass parameter, but it is not a suitable thermodynamic potential for
static-black-hole thermodynamics if $M$ is constant. It is also shown that the
Reverse Isoperimetric Inequality holds. We generalize the results to the case
of the Reissner-Nordstr\"{o}m-de Sitter black hole surrounded by quintessence
with two physical constants yielding two thermodynamic volumes.
