How can I reinforce an entertainment stand for an aquarium?
Disagree, aquariums do not hold most of their weight under the corners.  The weight is almost uniformly distributed on portions that are in contact with the supporting surface.  If it is an acrylic tank with no frame, then all that weight is distributed across the whole bottom surface (assuming it is resting on a shelf like you illustrate).  If it is a glass tank with a frame, then the weight is around the perimeter wherever the frame contacts the surface.

The rod under the corners are good, and your idea about diagonal members will prevent sagging.  But that is probably overkill and also adds complexity to the fabrication.  I would just run lengths of angle iron between the rods along the long length.  I don't think sag would be an issue on the short length.  

You didn't ask, but spend the extra money and get an acrylic tank (not just any plastic tank though, acrylic).  A glass tank will eventually leak especially if it is not on a perfectly flat surface (flat, not necessarily level), and sometimes even then.

Lastly, that entertainment center is probably made of MDF or a similar material.  It will swell like mad if left in contact with water long enough.  No matter what you do, you will end up with some water outside the tank at some point.  So, this is not the best material overall so do take care.