Something that's done hotly happens with anger or excitement. If you hotly debate your school's proposed dress code, it means you have a fierce, passionate opinion about the subject.
If your friend says she heard a rumor that you throw recyclable bottles in the trash, you'll hotly deny it, especially if you care deeply about the environment. And if your English teacher abruptly ends his tradition of pizza parties on the first Friday of every month, the class will hotly object. While this adverb has been around since the 16th century, it originally only referred to temperature, as in "the fire burned hotly."