  Considering the theory of induced gravity coupled to matter fields, taking
the $\phi ^6$ interaction potential model we evaluate the one-loop effective
potential in a (3+1)dimensional Bianchi type-I spacetime. It is proved that the
$\phi ^6$ theory can be regularised in (3+1)dimensional curved spacetime. We
evaluate the finite temperature effective potential and study the temperature
dependence of phase transitions. The nature of phase transitions in the early
universe is clarified to be of first order. The effects of spacetime curvature
and arbitrary field coupling on the phase transitions in the early universe are
also discussed.
