In "The Last Olympian" by Rick Riordan, the theme of hope is intricately tied to a sense of belonging and safety. The phrase "Hope survives best at the Hearth" suggests that hope flourishes in environments where love and support are present, reminiscent of the warmth and comfort associated with home. The hearth symbolizes a central place of warmth, family, and community, highlighting that connections with loved ones can nurture resilience and optimism, especially during challenging times.

The narrative emphasizes that in the face of adversity, the strength drawn from relationships and a supportive community can be a powerful source of hope. As characters navigate their trials, it's their bonds with each other that propel them forward. Riordan effectively communicates that, while challenges may be daunting, hope thrives in the presence of unity and love, underscoring the importance of cherishing those connections in our own lives.