We consider the thermodynamic properties of an exact black hole solution

obtained in Weyl geometric gravity theory, by considering the simplest

conformally invariant action, constructed from the square of the Weyl scalar,

and the strength of the Weyl vector only. The action is linearized in the Weyl

scalar by introducing an auxiliary scalar field, and thus it can be

reformulated as a scalar-vector-tensor theory in a Riemann space, in the

presence of a nonminimal coupling between the Ricci scalar and the scalar

field. In static spherical symmetry, this theory admits an exact black hole

solution, which generalizes the standard Schwarzschild-de Sitter solution

through the presence of two new terms in the metric, having a linear and a

quadratic dependence on the radial coordinate, respectively. The solution is

obtained by assuming that the Weyl vector has only a radial component. After

studying the locations of the event and cosmological horizons of the Weyl

geometric black hole, we investigate in detail the thermodynamical (quantum

properties) of this type of black holes, by considering the Hawking

temperature, the volume, the entropy, specific heat and the Helmholtz and Gibbs

energy functions on both the event and the cosmological horizons. The Weyl

geometric black holes have thermodynamic properties that clearly differentiate

them from similar solutions of other modified gravity theories. The obtained

results may lead to the possibility of a better understanding of the properties

of the black holes in alternative gravity, and of the relevance of the

thermodynamic aspects in black hole physics.

According to the article, the authors have examined the thermodynamic properties of an exact black hole solution in Weyl geometric gravity theory. They have used the simplest conformally invariant action, constructed from the square of the Weyl scalar and the strength of the Weyl vector. By linearizing the action in the Weyl scalar and introducing an auxiliary scalar field, the theory can be reformulated as a scalar-vector-tensor theory in a Riemann space with a nonminimal coupling between the Ricci scalar and the scalar field.

In static spherical symmetry, this theory gives rise to an exact black hole solution that generalizes the standard Schwarzschild-de Sitter solution. The metric of the black hole solution includes two new terms that have linear and quadratic dependencies on the radial coordinate.

The authors then investigate the thermodynamic properties of this type of black hole. They analyze the locations of the event and cosmological horizons of the Weyl geometric black hole and study the quantum properties by considering the Hawking temperature, volume, entropy, specific heat, and Helmholtz and Gibbs energy functions on both horizons.

They find that Weyl geometric black holes have distinct thermodynamic properties that differentiate them from similar solutions in other modified gravity theories. These results may contribute to a better understanding of black holes in alternative gravity theories and the importance of thermodynamic aspects in black hole physics.

## Future Roadmap

To further explore the implications of Weyl geometric gravity theory and its black hole solutions, future research can focus on:

- Extension to other geometries: Investigate whether the exact black hole solutions hold for other types of symmetries, such as rotating or more general spacetimes.
- Quantum aspects: Consider the quantum properties of Weyl geometric black holes in more detail, such as evaluating the quantum fluctuations and their effects on the thermodynamics.
- Comparison with observations: Study the observational consequences of Weyl geometric black holes and compare them with astrophysical data, such as gravitational wave signals or observations of black hole shadows.
- Generalizations and modifications: Explore possible generalizations or modifications of the Weyl geometric theory that could lead to new insights or more accurate descriptions of black holes.

## Potential Challenges

During the research and exploration of the future roadmap, some challenges that may arise include:

- Complexity of calculations: The calculations involved in studying the thermodynamic properties of black holes in Weyl geometric gravity theory can be mathematically complex. Researchers will need to develop precise techniques and numerical methods to handle these calculations reliably.
- Data availability: Obtaining accurate astrophysical data for comparison with theoretical predictions can be challenging. Researchers may need to depend on simulated data or future observations to test their theoretical models.
- New mathematical tools: Investigating alternative gravity theories often requires the development and application of new mathematical tools. Researchers may need to collaborate with mathematicians or utilize advanced mathematical techniques to address specific challenges.

## Potential Opportunities

Despite the challenges, there are potential opportunities for researchers exploring the thermodynamics of Weyl geometric black holes:

- New insights into black hole physics: The distinct thermodynamic properties of Weyl geometric black holes offer a unique perspective on black hole physics. By understanding these properties, researchers can gain new insights into the nature of black holes and their behavior in alternative gravity theories.
- Applications in cosmology: The study of black holes in alternative gravity theories like Weyl geometric gravity can have implications for broader cosmological models. Researchers may discover connections between black hole thermodynamics and the evolution of the universe.
- Interdisciplinary collaborations: Exploring the thermodynamics of Weyl geometric black holes requires expertise from various fields, including theoretical physics, mathematics, and astrophysics. Collaborations between researchers from different disciplines can lead to innovative approaches and solutions to research challenges.

In conclusion,the research presented in the article provides valuable insights into the thermodynamic properties of black hole solutions in Weyl geometric gravity theory. The future roadmap outlined here aims to further explore these properties, address potential challenges, and take advantage of the opportunities that arise from studying Weyl geometric black holes.