jacob_drori comments on Memorization-generalization in practice

[jacob_drori]

Vary temperature t and measure the resulting learning coefficient function $\lambda \left(t\right)$

This confuses me. IIUC, $p_{\beta ,n}\left(w\right)=\frac{\phi \left(w\right)\exp (-n\beta L_n(w\left)\right)}{\int d{w}^{\prime }\phi \left(w^{\prime }\right)\exp (-n\beta L_n(w^{\prime }\left)\right)}$. So changing temperature is equivalent to rescaling the loss by a constant. But such a rescaling doesn’t affect the LLC.

What did I misunderstand?