War3:Source

Intresting question, one way this should be possible is by hooking the weapon_fire event(as the name suggests, that event is invoked whenever a weapon is fired.. duh) and remember that this shot is going to have a 1.5 damage multipler. Like this:
1. player fires gun, weapon_fire event is invoked
2. weapon_fire event handler sets some global to remember current state(when player fired and if damage is modified)
3. when w3 damage event fires(if it fires after weapon_fire, which it should) check if we need to double damage
4. repeat! The next time the player fires his gun the code in weapon_fire will detect that his last shot was not 10 seconds ago and deny the damage modifier(or allow if it actually was >= 10 seconds ago)

some semi-pseudoish code:
bool g_bDamageMultipler[MAXPLAYERSCUSTOM]; //remember if we need to apply damage multipler
int g_iLastTimeFired[MAXPLAYERSCUSTOM]; //holds war3 delay tracker "object"

public void OnPluginStart() {
   HookEvent("weapon_fire", OnWeaponFire);
   for(int i=0; i<MAXPLAYERSCUSTOM; ++i)
      g_iLastTimeFired[i] = War3_RegisterDelayTracker();
}

// do this in OnRaceChange/OnRaceSelected too
public void OnClientPutInServer(int client) {
  g_bDamageMultipler[client] = false;
  g_iLastTimeFired[client] = 0;
}
public void OnRaceSelected(int client, int old,int new) {
  if(old == thisRaceID)
     OnClientPutInServer(client);
 // or something like that
}

public void OnWeaponFire(Event ev, const char[] name, bool dontBroadcast) {
   int client = GetClientOfUserId(ev.GetInt("userid"));
   // ready for damage modifier?
   if(g_bDamageMultipler[client] && War3_TrackDelayExpired(g_iLastTimeFired[client]) {
      // cooldown expired, set modifier
      g_bDamageMultipler[client] = true;
      // reset delay tracker
      War3_TrackDelay(g_iLastTimeFired[client], 10.0);
   } else {
     // not ready
     g_bDamageMultipler[client] = false;
   }
}

OnW3TakeDamageThing(client,victim) {
  if(g_bDamageMultipler[client]) {
    //do stuff
  }
}

but take care if you hook weapon_fire I think it had some problems when using the knife(event firing pretty often) so just check if !War3_IsMeleeWeapon and you should be good to go 

I'm not actually sure if this is the best way, maybe someone else has a better idea.