Here are some design notes when it comes to modern pergola designs. The most common pergola design is to have decorative and somewhat rounded ends on the beam. The easiest way of creating a more modern pergola is to do away with the decorative ends.
lighting /windows/styling - Doorway hidden in volcanic stone wall of house by PK Arkitektar
Image 4 of 15 from gallery of Bondi House / Fearns Studio. Photograph by Tom Ferguson
Tresarca Residence by Assemblage Studio / Las Vegas, Nevada, USA - day bed, screen and curtain, outdoor kitchen.
Pergolato - Stratheden House on The Bishops Avenue in north London. There are many styles of pergola. This modern pergola feels clean and unfussy.
Boxwood cabanas (Mondrian Miami) want this!