A B737-800 Captain reports about recurring multiple malfunctions involving the Engine Thrust Reverser System. The P5 aft overhead reverser light intermittently illuminates when throttles are advanced during taxiing or takeoff; causing low speed abort. The problem has also occurred after landing. The problem (warning) recurs even after the Engine Accessory Unit (EAU) is reset.

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Reporter stated a single light for each engine will illuminate 'REVERSER' on the P5 aft overhead panel when a mechanical malfunction occurs in the Thrust Reverser System. The engine annunciator light on the six-pack at the glare shield will also illuminate because that light is 'latched' to the reverser overhead light. A master caution will also illuminate. The reverser light on the P5 overhead panel is deferrable if the forward Common Display System (CDS) display unit also shows a REV message. Reporter stated he has noticed the overhead reverser light(s) have come on when the throttles are advanced; then retarded; without ever moving the reverser lever. This has happened when throttles are moved up for taxiing; takeoff; resulting in low speed abort and on landing. He has not experienced an in-flight reverser overhead light illuminating; but other pilots have told him they have had that happen to them; one Captain noted that both reverser lights came on his B737-800. Their Quick Reference Handbook (QRH) does not require them to divert when the overhead reverser light(s) come; but to continue to destination because the flight manual indicates that multiple malfunctions are required for an in-flight reverser deployment to occur.Reporter stated that if he gets a reverser light after engine start; he returns to the gate; gets Maintenance or Ramp Personnel to clear the area; pushes the engine throttle up; then retards the throttle; moves the reverser lever to deploy; holds the reverse position momentarily; then stows the reverser. The reverser light goes out and the problem goes away; as long as the affected engine is not shut down again. The reverser light problem occurred each time after he had shut down the engine and restarted them on three different flights on the same B737-800 he mentions in his report.Reporter stated that Maintenance performs the BITE test on the Engine Accessory Unit (EAU) and the unit is reset and signed-off; but the reverser light eventually comes back on. Because he is not getting any response from his carrier about the multiple malfunctions that show up on the EAU in the E/E compartment; including the recurring overhead light; he is refusing aircraft with reverser issues unless Maintenance Control directs Maintenance to lock-out and pin the reverser. He is also entering his refusals in the logbook; so that outbound pilots scheduled for the aircraft will be aware of the reverser issues on the aircraft.