On April 19, 2000, at 1545 eastern daylight time, a Piper PA-31-350, N41081, experienced an in-flight fire on the right engine while on approach to land at Branch Field in Rock Hill, South Carolina. The business flight was operated by the pilot under the provisions of Title 14 CFR Part 91 with an instrument flight plan filed. Visual weather conditions prevailed at the time of the accident. The airplane sustained substantial damage and the private pilot and his passenger were not injured. The flight departed Summerville, South Carolina, at 1500. According to the pilot, upon arriving at the destination airport, the pilot completed a landing check, which included lowering the landing gear. The pilot continued the approach and landed on runway 20. After landing, the pilot of another airplane waiting for takeoff radioed the pilot of the accident airplane and told him that there was a fire on the right engine. The pilot of the accident airplane confirmed the fire and shut down the right engine. The engine fire was subsequently extinguished after airport officials used several hand-held fire extinguishers. The external examination of the airplane revealed that the right engine cowling had sustained heavy fire damage to the aft half of the assembly. Cowling fire damage extended completely around the entire nacelle. The bottom side of the wing assembly was also fire damaged aft of the engine firewall to a point mid-chord of the right wing flap assembly. The internal examination of the engine compartment revealed fire damage in the vicinity of the engine driven fuel pump. Adjacent to the fuel pump assembly is a hydraulic supply line. Additional examination of the engine compartment disclosed that the hydraulic supply line was loose; the engine driven fuel pump was also loose. A review of the airframe and engine maintenance log books revealed that Airworthiness Directive (AD) 98-18-12 was required. The AD requires that the engine driven pump assembly be re-torqued "to prevent leaks which could result in engine failure or engine fire." The log book review failed to show compliance with this AD.