cp = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
      PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
      ColorFunction -> "Rainbow", ImageSize -> 500, 
      Axes -> True]; 

    Replace[cp, {Tooltip[{d___, l__Line}, t : (0. | 1. | -1.)] :> {Thick, Opacity[1], 
        t /. {0. -> Green, -1. -> Red, 1. -> Blue, _ -> {d}},  Tooltip[{l}, t]}}, All]

[![enter image description here][1]][1]

Alternatively, extract the contours from `cp` and redo `ContourPlot` using styled contours:

    contours = Cases[cp, Tooltip[_, t_] :> t, All];

    styledcontours = Join[ {#, Directive[##2]} & @@@ 
        Thread[{{0., -1., 1.}, {Green, Red, Blue}, Thick, Opacity[1]}], 
       Complement[contours, {0., -1., 1.}]];
    
    ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
     PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
     ColorFunction -> "Rainbow", ImageSize -> 500, Axes -> True, 
     Contours -> styledcontours]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/ymVMW.png
  [2]: https://i.sstatic.net/zrUtK.png