No, the finite speed of light is not evidence that the universe is hyperbolic. The finiteness of the speed of light is a fundamental property of the universe, which is consistent with the theory of special relativity. It does not directly imply anything about the curvature or shape of the universe.
The curvature of the universe is a topic studied in the field of cosmology, and it is related to the distribution of matter and energy in the universe. According to the current leading cosmological model, known as the Lambda-Cold Dark Matter (ΛCDM) model, the universe is assumed to have a nearly flat geometry. This means that on large scales, the curvature is very close to zero.
The concept of hyperbolic geometry relates to a different type of curvature that is intrinsic to the geometry of surfaces. In hyperbolic geometry, the sum of the angles in a triangle is less than 180 degrees, unlike in Euclidean (flat) geometry where the sum is exactly 180 degrees. Hyperbolic geometry has its own unique mathematical properties and is not directly related to the finite speed of light or the curvature of the universe.
The question of the precise curvature of the universe is an active area of research, and scientists continue to gather data and study various cosmological observations to refine our understanding of the universe's geometry. However, current observational evidence suggests that the universe is very close to flat based on data from cosmic microwave background radiation and other cosmological measurements.