We give an example of a non compact riemannian manifold with finite volume for which the limit corresponding to the clas-sical definition of the volumetric entropy does not exist. This confirms the fact that in the non compact finite volume case,the natural definition is given by the critical exponent of the mean growth rate for the volume on the riemannian covering. Subject classification AMS 2000 : Primary 37A35 ; Secondary : 37D40, 53C24