In a small bowl, combine the warm water, sugar, and active dry yeast. Stir gently and let it sit for 5 to 10 minutes until the mixture becomes frothy.
In a large mixing bowl, whisk together the all-purpose flour and fine sea salt.
Make a well in the center of the dry ingredients and pour in the yeast mixture and 1/4 cup olive oil.
Using a wooden spoon or your hands, mix until a sticky dough forms.
Turn the dough onto a lightly floured surface and knead for about 8 to 10 minutes until smooth and elastic. Add small amounts of flour if the dough is too sticky, but keep it soft.
Lightly oil a large bowl with olive oil. Place the dough in the bowl and turn it to coat with oil. Cover the bowl with a clean kitchen towel or plastic wrap and let it rise in a warm place for 1 to 1 1/2 hours, or until doubled in size.
Preheat your oven to 425°F (220°C).
Generously oil a 9×13-inch (23×33 cm) baking pan with olive oil.
Punch down the risen dough and transfer it to the prepared pan. Stretch and press the dough to fit the pan evenly.
Cover the dough again and let it rest for 20 minutes to puff slightly.
Using your fingertips, dimple the surface of the dough all over, pressing down about 1/2 inch deep.
Drizzle 2 tablespoons of olive oil evenly over the top of the dough.
Sprinkle the chopped fresh rosemary and coarse sea salt evenly on top.
Bake the focaccia in the preheated oven for 20 to 25 minutes, or until golden brown and cooked through.
Remove from the oven and let it cool in the pan for 10 minutes before transferring to a wire rack.
Slice into 8 pieces and serve warm or at room temperature.