So this legendary

[Denona-khazgoroth (Denona)]

@Aguon-hellscream

Does it have a higher drop chance on harder difficulties?

Yes

@Linxy

which will be very rare on Raid Finder difficulty (when available) and significantly increases in drop rate with each difficulty.

@Aguon-hellscream

And is it likenthe love rocket mount, that is, the first kill yields a higher drop chance?

No. The blp is personal to that character. So killing it on alts will not increase the chances on your main.