You can store your buffer at room temperature, or even at 4 degree, just as sugested by Feng Lin. Bear in mind that when using Tris as a buffering component, the pH of your buffer will be different when stored at 4 degree or RT. Since pH plays a crucial role in maintaining the stability of proteins, therefore, it is important for you to measure the pH accordingly, when storing the buffer either at 4 degree or at RT.
You may want to refer to this site for more information:
I agree, 4 degrees is the way to go. I'm assuming you keep your samples on ice when doing the extraction, so it only makes sense to use pre-chilled extraction buffer.
The storage buffer(without the protein) can be stored at room temperature. But the point is when you want to use it gain after a while you have to re add PMSF and Benzenimide again since they go bad after a while.
The best suitable temperature to store protein in buffer is 4 Degree Celsius. But, you can store buffer at room temperature (Only Extraction Buffer without Protein).