  Given a topological orientable surface of finite or infinite type equipped
with a pair of pants decomposition $\mathcal{P}$ and given a base complex
structure $X$ on $S$, there is an associated deformation space of complex
structures on $S$, which we call the Fenchel-Nielsen Teichm\"uller space
associated to the pair $(\mathcal{P},X)$. This space carries a metric, which we
call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We
studied this metric in the papers \cite{ALPSS}, \cite{various} and
\cite{local}, and we compared it to the classical Teichm\"uller metric (defined
using quasi-conformal mappings) and to another metric, namely, the length
spectrum, defined using ratios of hyperbolic lengths of simple closed curves
metric. In the present paper, we show that under a change of pair of pants
decomposition, the identity map between the corresponding Fenchel-Nielsen
metrics is not necessarily bi-Lipschitz. The results complement results
obtained in the previous papers and they show that these previous results are
optimal.
