  We study $d$-dimensional $AdS$ black holes surrounded by Dark Energy (DE),
embedded in $D$-dimensional M-theory/superstring inspired models having $AdS_d
\times \mathbb{S}^{d+k}$ space-time with $D=2d+k$. We focus on the
thermodynamic Hawking-Page phase transitions of quintessential DE black hole
solutions, whose microscopical origin is linked to $N$ coincident
$(d-2)$-branes supposed to live in such $(2d+k)$-dimensional models.
Interpreting the cosmological constant as the number of colors $\propto
N^{\frac{d-1}{2}}$, we calculate various thermodynamical quantities in terms of
brane number, entropy and DE contributions. Computing the chemical potential
conjugated to the number of colors in the absence of DE, we show that a generic
black hole is more stable for a larger number of branes for lower dimensions
$d$.In the presence of DE, we find that the DE state parameter $\omega_q$
should take particular values, for $(D,d,k)$ models, providing a non trivial
phase transition structure.
