Comparisons of Lie algebra cohomologies of $(\varphi,\Gamma)$-modules

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between $\mathbb{Q}_p$ and $L$-analytic Lie algebra cohomology of analytic $(\varphi_L,\Gamma_L)$-modules. We use the results to conclude that for $L\neq \mathbb{Q}_p,$ there exist examples of \'etale $(\varphi_L,\Gamma_L)$-modules over Robba rings whose $\mathbb{Q}_p$-analytic cohomology does not arise as a base change of Galois cohomology.