The drop rates are greatly increased with the difficulty, here are the applied mmodifiers :
(note that each mob has a n base % to drop a legendary, if the proc is for example 10%, playing T6 will raise the probability to 23% instead, it's not guaranteed drop in any way, just better probability to have one. )
Torment I: 819% Health, 396% Damage, 300% extra gold bonus, 300% extra XP bonus,[1] 15% increased legendary drop rate
Torment II: 1311% Health, 575% Damage, 400% extra gold bonus, 400% extra XP bonus, 32% increased legendary drop rate
Torment III: 2097% Health, 833% Damage, 550% extra gold bonus, 550% extra XP bonus, 52% increased legendary drop rate
Torment IV: 3355% Health, 1208% Damage, 800% extra gold bonus, 800% extra XP bonus, 74% increased legendary drop rate
Torment V: 5369% Health, 1752% Damage, 1150% extra gold bonus, 1150% extra XP bonus, 101% increased legendary drop rate
Torment VI: 8590% Health, 2540% Damage, 1600% extra gold bonus, 1600% extra XP bonus, 131% increased legendary drop rate