I have brown hot water, dark at first then getting lighter after running for awhile. I have city water and cold is never discolored. I also have a water softener. The house is only 4 years old and this has been happening for 2 or more years. I had it flushed by plumber last year and it was better for awhile. I've talked to company that put the heater it in and they said I need a new anode rod and chlorinated, but it would only help for awhile due to having softener. What should I do? Would anode rod and labor be covered by warranty? T

It sounds like there is sediment in your hot water heater. Cold water isn't ran through the hot water tank, so it wouldn't be discolored in this scenario, but the hot water would be. You would have to check with the company to see if the Anode rod and labor would be covered by the company.
http://www.rd.com/images/tfhimport/2006/200609_WaterHeater_002.jpg
Here is an image of an anode rod that is new, and old. As you can see, this could be causing the discoloration.

Your brown water is caused by iron oxide,(rust) which can be built up on anode rod as Carl suggests or from mineral buildup if you don't flush your heater on a regular maintenance schedule.
For long life and fewer troubles you should keep your heater clear of mineral build-up by FLUSHING NOT DRAINING on a regular schedule. Attach a hose to the boiler drain at the bottom of the tank. With the pressure on, open the boiler drain and let it run until the water runs clear. You will see a spurt of red,(rust) followed by white or yellow grains,(lime or calcium carbonate). This shouldn't take more then a few minutes. Do this monthly to keep it clear. Now flush out your hot water lines on ALL fixtures that are affected . Now pull each aerator and clean the screens. Be sure you put them back together the same way you took them out. Don't forget to flush it out every month. Your heater will thank you for it. Hope this helps and thank you for rating my answer, Tom

I agree with Tom on maintaining your water heater and the probable cause. In addition, check that your water softener is providing soft water to your cold as well, with the exception of the kitchen sink and outside faucets. If it is only going to the hot try bypassing your softener for a while after flushing out your heater.
And to further answer your question, the anode rod is probably still under the manufacturers warranty but labor is not included in that. You could call the manufacturer with your model number and serial number to find out.

Are you getting this brown water in all you hot water fixtures? And how long do you run water for it to clear up?