The term is "enjo kosai", usually translated as compensated dating.
It runs the gamut from the concept for Rent-A-Girfriend all the way to prostitution in everything but name, but usually it's intended for men (and women) to pay for temporary companionship.

Intimacy is unfortunately very hard to find in Japan. When it comes to making ends meets versus finding love, most people will always stick to the former option due to the pressure that society puts on you to working your ass off til you die. Without any love in a person's life, lacking something even as small as a hug or a warm smile to start the day will drive anyone insane.