  We consider the two-dimensional random-phase sine-Gordon and study the
vicinity of its glass transition temperature $T_c$, in an expansion in small
$\tau=(T_c-T)/T_c$, where $T$ denotes the temperature. We derive
renormalization group equations in cubic order in the anharmonicity, and show
that they contain two universal invariants. Using them we obtain that the
correlation function in the super-rough phase for temperature $T<T_c$ behaves
at large distances as $\bar{<[\theta(x)-\theta(0)]^2>} =
\mathcal{A}\ln^2(|x|/a) + \mathcal{O}[\ln(|x|/a)]$, where the amplitude
$\mathcal{A}$ is a universal function of temperature
$\mathcal{A}=2\tau^2-2\tau^3+\mathcal{O}(\tau^4)$. This result differs at
two-loop order, i.e., $\mathcal{O}(\tau^3)$, from the prediction based on
results from the "nearly conformal" field theory of a related fermion model. We
also obtain the correction-to-scaling exponent.
