  The ongoing observations of merging black holes by the instruments of the
fledging gravitational wave astronomy has opened the way for testing the
general relativistic Kerr black hole metric and, at the same time, for probing
the existence of more speculative horizonless ultracompact objects. In this
paper we quantify the difference that these two classes of objects may exhibit
in the post-merger ringdown signal. By considering rotating systems in general
relativity and assuming an eikonal limit and a third-order Hartle-Thorne slow
rotation approximation, we provide the first calculation of the early ringdown
frequency and damping time as a function of the body's multipolar structure.
Using the example of a gravastar, we show that the main ringdown signal may
differ as much as a few percent with respect to that of a Kerr black hole, a
deviation that could be probed by near future Advanced LIGO/Virgo searches.
