  We study two-component Bose gases with Raman induced spin-orbit coupling via
a perturbation approach at finite temperature. For weak coupling, free energy
is expanded in terms of Raman coupling strength up to the second order, where
the coefficient (referred to as Raman susceptibility) is determined according
to linear response theory. The equation of state for the stripe phase and the
plane-wave phase are obtained in Popov approximation, and the first order
transition between these two phases is investigated. As temperature increases,
we find the phase boundary bends toward the stripe phase side in most
temperature regions, which implies the ferromagnetic order is more robust than
the crystalline order in presence of thermal fluctuations. Our results
qualitatively agree with the recent experimental observation in rubidium atomic
gases. A method to measure the Raman susceptibility through the two-photon
Bragg scattering experiment is also discussed.
