Cosmological consequences of Yukawa-unified SUSY with mixed axion/axino cold and warm dark matter

27 pages including 16 EPS figuresSupersymmetric models with t-b-\tau Yukawa unification at M_{GUT} qualitatively predict a sparticle mass spectrum including first and second generation scalars at the 3--15 TeV scale, third generation scalars at the (few) TeV scale and gluinos in the sub-TeV range. The neutralino relic density in these models typically turns out to lie far above the measured dark matter abundance, prompting the suggestion that instead dark matter is composed of an axion/axino mixture. We explore the axion and thermal and non-thermal axino dark matter abundance in Yukawa-unified SUSY models. We find in this scenario that {\it i}). rather large values of Peccei-Quinn symmetry breaking scale f_a\sim 10^{12} GeV are favored and {\it ii}). rather large values of GUT scale scalar masses \sim 10-15 TeV allow for the re-heat temperature T_R of the universe to be T_R\agt 10^6 GeV. This allows in turn a solution to the gravitino/Big Bang Nucleosynthesis problem while also allowing for baryogenesis via non-thermal leptogenesis. The large scalar masses for Yukawa-unified models are also favored by data on b\to s\gamma and B_s\to \mu^+\mu^- decay. Testable consequences from this scenario include a variety of robust LHC signatures, a possible axion detection at axion search experiments, but null results from direct and indirect WIMP search experiments