  Multi-spinning higher dimensional Kerr-AdS black holes admit the stable
small-large black hole phase transition of van der Waals type. In this paper,
we study the exact critical phenomena and phase structure in five-dimensional
spacetime. First, we examine the thermodynamic laws in the reduced parameter
space and find that they are quite different from the conventional
thermodynamic laws. Then based on the reduced laws, the phase structure in
different parameter spaces is investigated. The stable and metastable black
hole phases are clearly displayed. We present a highly accurate fitting formula
for the coexistence curve of small and large black hole phases. Using this
fitting formula, we examine the critical exponents when the black hole system
approaches the critical point along the coexistence curve. Moreover, employing
the dimensional analysis and symmetry analysis, we also give a numerical study
of the critical point for the unequal spinning black holes. These results are
very useful on further understanding the microstructure of the multi-spinning
black holes in higher dimensional spacetime.
