For the rosemary simple syrup: Combine the sugar, rosemary and 2 cups water in a medium saucepan over medium heat. Cook, stirring occasionally, until the sugar has dissolved. Remove from the heat and let the rosemary infuse for 20 minutes. Strain the syrup into a container, discarding the rosemary. Let cool to room temperature.
For the rosemary greyhound: Place the grapefruit juice, vodka and 1 ounce rosemary simple syrup in a cocktail shaker with ice. Shake well to mix. Serve on the rocks, garnished with a grapefruit slice.