cold comfort(n = noun.feeling) - very limited consolation or empathy; "he told me that time heals all wounds but that was cold comfort to me"
is a kind of consolation, solace, solacement

N  discontent, discontentment, dissatisfaction, dissent, disappointment, mortification, cold comfort, regret, repining, taking on, heart-burning, heart-grief, querulousness, hypercriticism, inquietude, vexation of spirit, soreness, worry, concern, fear, malcontent, grumbler, growler, croaker, dissident, dissenter, laudator temporis acti, censurer, complainer, fault-finder, murmerer, cave of Adullam, indignation meeting, winter of our discontent, with what I most enjoy contented least, discontented, dissatisfied, unsatisfied, ungratified, dissident, dissentient, malcontent, malcontented, exigent, exacting, hypercritical, repining, regretful, down in the mouth, in high dudgeon, in a fume, in the sulks, in the dumps, in bad humor, glum, sulky, sour as a crab, soured, sore, out of humor, out of temper, disappointing, unsatisfactory, frustrated (failure), Int, so much the worse!, that won't do, that will never do, it will never do, curtae nescio quid semper abest rei, ne Jupiter Quidem omnibus placet, poor in abundance, famished at a feast.

blow hot and cold, catch cold, cold, cold blooded, cold cash, cold cereal, cold chisel, cold cream, cold cuts, cold duck, cold feet, cold fish, cold frame, cold front, cold fusion, cold gangrene, cold medicine, cold rubber, cold shoulder, cold snap, cold sober, cold sore, cold spell, cold storage, cold stuffed tomato, cold sweat, cold turkey, cold type, cold war, cold water, cold wave, cold weather, cold work, comfort, comfort food, comfort station, comfort woman, comfort zone, common cold, creature comfort, head cold, ice cold, in cold blood, knock cold, louis comfort tiffany, out cold, pour cold water on, stone cold, throw cold water on