Which deep learning framework have support for gtx580 GPU?
Unfortunately you can't use GTX 580+Tensorflow.

TensorFlow GPU support requires having a GPU card with NVidia Compute Capability >= 3.0, while GTX 580 has only 2.0 compute capability.

However, caffe supports CUDA compute capability 2.0. You can have a try.

References

https://www.tensorflow.org/versions/r0.10/get_started/os_setup#prepare_environment_for_linux

https://developer.nvidia.com/cuda-gpus