Comparisons of Lie algebra cohomologies of $(\phi ,\mathrm{\Gamma })$-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 $({\phi }_L,{\mathrm{\Gamma }}_L)$-modules. We use the results to conclude that for $L\mathrm{\ne }{\mathbb{Q}}_p,$ there exist examples of étale $({\phi }_L,{\mathrm{\Gamma }}_L)$-modules over Robba rings whose ${\mathbb{Q}}_p$-analytic cohomology does not arise as a base change of Galois cohomology.