Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions

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.