Combine the sugar, lemon zest and juice in a medium saucepan over medium-high heat. Add the blackberries and cook until the berries release their juices, 6 to 7 minutes depending on how ripe the berries are. Continue to cook until the sauce has thickened slightly but the berries are still somewhat whole. Serve immediately as a warm compote or cold as a chilled compote. To serve cold, cool to room temperature about 10 minutes, then refrigerate, covered.