Most rubber roofs are EPDM. I would suggest using a power washer with a 40 degree or wider nozzle tip, and start out spraying about 18" away from the roof membrane itself to see if that removes the algae or fungus (most likely it's algae). Be careful if you are not familiar with using a power sprayer/washer, as they generate a high psi and can cut through rubber if you get too close. A mild dish washing detergent in a bucket of warm water with a soft bristle brush is another alternative if you don't have a power washer. Be certain to rinse very well to remove any soap residue.
The idea of contacting a commercial roofing company that installs EPDM material is the best starting point. I would not be putting anything on that roof that could puncture the membrane, as most EPDM roofs are not intended to be walked on. They will get soft just from the Summer heat, even if the sun does not shine directly on it. In commercial flat EPDM roofs, a ballast of small, round river rock is usually applied to keep the membrane from being lifted in high winds, and specialized walking materials are applied to pathways about the roof for servicing roof top equipment such as HVAC units and exhaust fans.
