We point out that the spectral geometry of hyperbolic manifolds provides a remarkably precise model of the modern conformal bootstrap. As an application, we use conformal bootstrap techniques to derive rigorous computer-assisted upper bounds on the lowest positive eigenvalue $\lambda_1(X)$ of the Laplace-Beltrami operator on closed hyperbolic surfaces and 2-orbifolds $X$. In a number of notable cases, our bounds are nearly saturated by known surfaces and orbifolds. For instance, our bound on all genus-2 surfaces $X$ is $\lambda_1(X)\leq 3.8388976481$, while the Bolza surface has $\lambda_1(X)\approx 3.838887258$. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case.