In a medium sauce pan over medium heat, heat oil. Add garlic and cayenne and cook until fragrant, 1 minute.
Add cherry tomatoes, 1/4 cup cold water, vinegar, honey, and rosemary. Season with salt and pepper and bring to a boil. Reduce heat to medium-low and simmer, stirring occasionally, until mixture is thick and jammy, 22 to 24 minutes. Remove from heat and season again with salt and pepper if needed.
Remove rosemary and transfer jam into a glass jar. Cool completely and refrigerate.