Since the use text on the beating sludge heart implies that the heart is in 'pain' (although this is definitely arguable), it would be fairly reasonable for it to trigger cool coin or born of sorrow.
Storywise, it can swing either way. Mechanics-wise, I see a couple factors. One is it could potentially get interesting to have a surefire way to get a cold coin effect without spending Energy — either if there's (eventually?) something specifically related to the effect, or if (this is a stretch) acquiring/'pulling' a beating sludge heart turns out to be a useful strategy to get fire defense and etheric power. :P
On the flip side, Kinak might not be happy with a surefire way to get a cold coin effect, particularly if you have multiple hearts and can have it on permanently. Even more so with born of sorrow (we still don't know if those have the same 'morality', correct?). A somewhat artificial 'only triggers once per day' could alleviate this, I suppose. ('Can only use all your hearts at once' is also artificial and allows mall trickery.)
It may well be more trouble than it's worth… but I felt the suggestions forum was getting neglected. Feel free to add suggestions for what else could potentially be added to the list of evil acts. :)