The stunning double homicide of “American Idol” music supervisor Robin Kaye and her husband has rocked their Los Angeles neighborhood amid skyrocketing break-ins in the area … and now the family directly next door wants to get the hell out of there.

The neighbor of the slain couple tells TMZ … he and his family are completely terrified — but not as shocked as they could be, since he says crime — home break-ins, specifically — in the Encino area has escalated dramatically the past few years.

We’re told, immediately after news broke of the killings, several neighbors had an impromptu meeting where they agreed to look out for one another … ’cause it was simply getting too dangerous to rely on home security systems alone.

As we reported Tuesday … 22-year-old Raymond Boodarian was arrested on suspicion of double homicide. LAPD detectives believe he has no relationship to Robin and her husband … and authorities say this was an isolated incident. Boodarian is believed to have entered the house through an unlocked door last Thursday afternoon, police said … and the couple came home and discovered him in their house, and they were shot in the confrontation. Their bodies were found on Monday.

The neighbor’s concerns seem to track, as TMZ was told by authorities they were called out to the property on Thursday when a person was seen by a neighbor scaling the couple’s fence … police searched the property and the area, but no suspects were located.

And even before that … we’re told there was a call for service to the slain couple’s home in the middle of the night on May 11 for as many as 5 people trespassing. It’s reported the couple looked into private security not long after.

The next-door neighbor tells us he did not know the couple well — they only ever exchanged pleasantries … the most interaction they ever had was when his hot tub was leaking underground and water began seeping onto the couple’s property.