We study phases of five-dimensional three-charge black holes with a
circle in their transverse space. In particular, when the black hole
is localized on the circle we compute the corrections to the metric
and corresponding thermodynamics in the limit of small mass. When
taking the near-extremal limit, this gives the corrections to the
finite entropy of the extremal three-charge black hole as a function
of the energy above extremality. For the partial extremal limit with
two charges sent to infinity and one finite we show that the first
correction to the entropy is in agreement with the microscopic
entropy by taking into account that the number of branes shift as a
consequence of the interactions across the transverse circle.
