The first delay came after the ninth over with the Royals on 85-2.
When play resumed, Jason Roy went on to score 61 off 42 balls (eight fours and three sixes) before falling to Romario Shepherd.
When the rain came again, the Royals were 131-4 after 15 overs with David Miller and Evan Jones at the crease.
The match was later called off, with both teams getting two log points.