  We calculate the explicit expression of the effective potential in a
$\lambda\phi^4$ theory at finite temperature in a static universe for arbitrary
spacetime dimensions (2\leq D < 5). To study the combined effects of the
temperature and scale factor to the spontaneous symmetry breaking we evaluate
the effective potential at finite temperature in $R\otimes S^{D-1}$ and
$R\otimes H^{D-1}$. The phase structure of a $\lambda\phi^4$ theory is found by
observing the minimum of the effective potential with varying temperature and
scale factor. All the ring diagrams are resummed for $D\gtrsim 4$ to improve
the loop expansion at high temperature. For a conformally coupled and a
minimally coupled scalar field it is shown that temperature and positive
curvature suppress the symmetry breaking, while negative curvature enhances it.
The conformally coupled scalar has larger curvature effects than the minimally
coupled one.
