We present FLAMO, a Frequency-sampling Library for Audio-Module Optimization designed to implement and optimize differentiable linear time-invariant audio systems. The library is open-source and built on the frequency-sampling filter design method, allowing for the creation of differentiable modules that can be used stand-alone or within the computation graph of neural networks, simplifying the development of differentiable audio systems. It includes predefined filtering modules and auxiliary classes for constructing, training, and logging the optimized systems, all accessible through an intuitive interface. Practical application of these modules is demonstrated through two case studies: the optimization of an artificial reverberator and an active acoustics system for improved response coloration.