  We construct a class of five dimensional black hole solutions to cubic
quasi-topological gravity with conformal scalar hair and study their
thermodynamics. We find these black holes provide the second example of black
hole $\lambda$-lines: a line of second order (continuous) phase transitions,
akin to the fluid/superfluid transition of $^4$He. Examples of isolated
critical points are found for spherical black holes, marking the first in the
literature to date. We also find various novel and interesting phase
structures, including an isolated critical point occurring in conjunction with
a double reentrant phase transition. The AdS vacua of the theory are studied,
finding ghost-free configurations where the scalar field takes on a non-zero
constant value, in notable contrast to the five dimensional Lovelock case.
