  We investigated the effect of Cd and Hg doping on the first order
superconducting (SC) transition and the high field-low temperature SC state of
CeCoIn$_5$ by measuring the specific heat of CeCo(In$_{\rm 1-x}$Cd$_{\rm
x}$)$_5$ with x=0.0011, 0.0022 and 0.0033 and CeCo(In$_{\rm 1-x}$Hg$_{\rm
x}$)$_5$ with x=0.00016, 0.00032, and 0.00048 at temperatures down to 0.1 K and
fields up to 14 T. Cd substitution rapidly suppresses the cross-over
temperature $T_{\rm 0}$, where the superconducting transition changes from
second to first order, to $T$=0 K with x=0.0022 for $H\parallel$ [100], while
it remains roughly constant up to x=0.0033 for $H\parallel$ [001]. The
associated anomaly of the proposed FFLO state in Hg-doped samples is washed out
by x=0.00048, while remaining at the same temperature, indicating high
sensitivity of that state to impurities. We interpret these results as
supporting the non-magnetic, possibly FFLO, origin of the high field - low
temperature state in CeCoIn$_5$.
