  We investigate the thermal physics of a Bose-Hubbard model with Rashba
spin-orbit coupling starting from a strong coupling mean-field ground state.
The essential role of the spin-orbit coupling $\left(\gamma\right)$ is to
promote condensation of the bosons at a finite wavevector ${k}_{0}$. We find
that the bosons display either homogeneous or phase-twisted or orbital ordered
superfluid phases, depending on $\gamma$ and the inter-species interaction
strength $\lambda$. We show that an increase of $\gamma$ leads to suppression
of the critical interaction $U_c$ for the superfluid to Mott insulator
transition in the ground state, and a reduction of the $T_c$ for superfluid to
Bose-liquid transition at a fixed interaction. We capture the thermal
broadening in the momentum distribution function, and the real space profiles
of the thermally disordered magnetic textures, including their homogenization
for $T \gtrsim T_{c}$. We provide a Landau theory based description of the
ground state phase boundaries and thermal transition scales, and discuss
experiments which can test our theory.
