  We study the Gregory-Laflamme instability of a large uniform black brane
wrapping a two-sphere compactification manifold. This paper extends the work
arXiv:hep-th/0604015, where the compactifications on p-torus were considered.
The new features of the spherical case are the non-zero curvature of the
compactification manifold and the absence of the rescaling symmetry due to a
built-in stabilization mechanism. We calculate the order of the phase
transition in dependence on the number d of extended dimensions using the
Landau-Ginzburg approach. It is found that for d > 11 a uniform spherical black
brane in microcanonical ensemble exhibits a smooth second order phase
transition towards a stable branch of non-uniform black brane solutions. The
critical number of extended dimensions, for which there is a change in the
order of the phase transition, is different for microcanonical and canonical
ensembles and does not coincide with the critical number of dimensions in the
case of the flat toric compactifications. We briefly discuss the origin of this
mismatch in the orders of phase transition for the different ensembles.

High Energy Physics - Theory;General Relativity and Quantum Cosmology