  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.
