  Holographic CFTs and holographic RG flows on space-time manifolds which are
$d$-dimensional products of spheres are investigated. On the gravity side, this
corresponds to Einstein-dilaton gravity on an asymptotically $AdS_{d+1}$
geometry, foliated by a product of spheres. We focus on holographic theories on
$S^2\times S^2$, we show that the only regular five-dimensional bulk geometries
have an IR endpoint where one of the sphere shrinks to zero size, while the
other remains finite. In the $Z_2$-symmetric limit, where the two spheres have
the same UV radii, we show the existence of a infinite discrete set of regular
solutions, satisfying an Efimov-like discrete scaling. The $Z_2$-symmetric
solution in which both spheres shrink to zero at the endpoint is singular,
whereas the solution with lowest free energy is regular and breaks $Z_2$
symmetry spontaneously. We explain this phenomenon analytically by identifying
an unstable mode in the bulk around the would-be $Z_2$-symmetric solution. The
space of theories have two branches that are connected by a conifold transition
in the bulk, which is regular and correspond to a quantum first order
transition. Our results also imply that $AdS_5$ does not admit a regular
slicing by $S^2\times S^2$.
