In a large bowl, combine the warm water, honey, and active dry yeast. Stir gently and let it sit for about 5 to 10 minutes. You will know it is ready when a foam forms on the surface, indicating the yeast is active.
Add the flour, fine sea salt, and two tablespoons of olive oil to the yeast mixture. Stir with a wooden spoon until a shaggy dough forms. You can use a stand mixer with a dough hook on low speed if you prefer. The dough should be quite sticky.
Drizzle a little olive oil into a clean bowl and place the dough inside. Cover the bowl with a damp cloth or plastic wrap. Place it in a warm, draft-free spot for about 1 to 1.5 hours, or until the dough has doubled in size.
Pour about 2 tablespoons of olive oil into a 9×13 inch baking pan. Transfer the risen dough to the pan. Gently stretch it toward the corners. If it shrinks back, let it rest for 10 minutes and try again. Cover and let it rise again for 30 to 45 minutes.
Preheat your oven to 425°F (220°C). Use your fingertips to press deep dimples all over the surface of the dough. Drizzle the remaining olive oil over the top. Press the halved cherry tomatoes and sliced garlic into the dimples. Sprinkle with fresh rosemary and flaky sea salt.
Place the pan in the oven and bake for 20 to 25 minutes. The focaccia is done when the top is golden brown and the tomatoes have started to blister and caramelize. Remove from the oven and let it cool in the pan for 10 minutes before slicing.