f[a_?NumberQ, b_?NumericQ] :=
  a + b - 3 Max[Select[Range[0, a, 1], b - # >= -(a - #)^2 &]]

RegionPlot3D[
  T >= -z^2 && p >= f[z, T]&& z>=0 , {T, -20, 20}, {z, -20, 20}, {p, -20, 20},
  Evaluated -> True, PlotRange -> Full]

Clear[f]
f[a_?NumberQ, b_?NumericQ] :=
  a + b - 3 Max[Select[Range[0, a, 1], b - # >= -(a - #)^2 &]]

RegionPlot3D[T >= -z^2 && p >= f[z, T] && z >= 0,  {T, -20, 20}, {z, -20, 20}, {p, -20, 20},
 PlotPoints -> 50, PlotRange -> All]