  We report specific heat measurements of the heavy fermion superconductor
CeCoIn5 in the vicinity of the superconducting critical field H_{c2}, with
magnetic field in the [110], [100], and [001] directions, and at temperatures
down to 50 mK. The superconducting phase transition changes from second to
first order for field above 10 T for H || [110] and H || [100]. In the same
range of magnetic field we observe a second specific heat anomaly within the
superconducting state. We interpret this anomaly as a signature of a
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. We
obtain similar results for H || [001], with FFLO state occupying a smaller part
of the phase diagram.
