  We study the properties of t-t'-V model of hard-core bosons on the triangular
lattice that can be realized in optical lattices. By mapping to the spin-1/2
XXZ model in a field, we determine the phase diagram of the t-V model where the
supersolid characterized by the ordering pattern (x,x,-2x') ("ferrimagnetic" or
SS A) is a ground state for chemical potential \mu >3V. By turning on either
temperature or t' at half-filling \mu =3V, we find a first order transition
from SS A to the elusive supersolid characterized by the (x,-x,0) ordering
pattern ("antiferromagnetic" or SS C). In addition, we find a large region
where a superfluid phase becomes a solid upon raising temperature at fixed
chemical potential. This is an analog of the Pomeranchuk effect driven by the
large entropic effects associated with geometric frustration on the triangular
lattice.
