Effective Potential for $\lambda\phi^4$ Theory at Finite Temperature in $R\otimes S^{D-1}$ and $R\otimes H^{D-1}$

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.