Is the analytical dependence of the theoretical Higgs potential with temperature known? At the very beginning of the unvierse, the Higgs potential had a paraboloid shape.
After the electroweak Symmetry Breaking, it took the mexican hat shape.
Is the analytical dependence of Higgs potential with temperature known ?
 A: The formalism used to describe the temperature dependence of the Higgs VEV is called a finite temperature effective potential (FTEP). 
The temperature contribution to the effective potential was calculated by Weinberg and by Jackiw & Dolan independently in 1974. The FTEP at one-loop order and at high temperature ($T\gg T_c$) looks like
$$
V_{\mathrm{T}}(\phi_{\mathrm{cl}})= V_{\mathrm{eff}}(\phi_{\mathrm{cl}}) + \frac{\lambda}{8}T^2\phi_{\mathrm{cl}}^2 - \frac{\pi^2}{90}T^4 + \cdots
$$
where $V_{\mathrm{eff}}(\phi_{\mathrm{cl}})$ is the zero-temperature one-loop effective potential (which will have the Mexican hat shape). This means that the additional $ \phi_{\mathrm{cl}}^2$ term due to the temperature contribution will dominate over the original $ \phi_{\mathrm{cl}}^2$ term due to the "mass term" in the Higgs potential. Here is a plot of the FTEP temperature dependence:

Note that the expression above is only valid for high temperatures, so plotting out that potential will not produce the same figure.
