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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 1−x Cd x ) 5 with x=0.0011, 0.0022 and 0.0033 and CeCo(In 1−x Hg 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 0 , where the superconducting transition changes from second to first order, to T =0 K with x=0.0022 for H [100], while it remains roughly constant up to x=0.0033 for H [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 .

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