Edit in JSFiddle

final int BOX_SIZE = 100;
PImage tex;

// フレームごとにテクスチャを更新する
void updateTexture() {
  colorMode(HSB, 360, 100, 100);

  // 都度 createImage をしないとテクスチャがアニメーションしない
  tex = createImage(0x80, 0x80, HSB);
  
  tex.loadPixels();
  for (int i = 0; i < tex.width; i++) {
    for (int j = 0; j < tex.height; j++) {
      int h = frameCount % 360;
      tex.set(i, j,
        (floor(i / 0x20) + floor(j / 0x20)) % 2 == 0 ?
        color(h, 10, 100) : color(h, 30, 50));
    }
  }
  tex.updatePixels();
  colorMode(RGB, 0xff, 0xff, 0xff);
}

void setup() {
  size(200, 200, P3D);
  tex = createImage(0x80, 0x80, HSB);
}

void draw() {
  background(0xff);

  camera();

  noStroke();
  translate(.5 * width, .5 * height);

  float angle = radians(millis() * 0.015);

  rotateZ(angle);
  rotateX(angle);
  rotateY(angle);

  updateTexture();
  drawBox(BOX_SIZE);

}

void drawBox(float boxSize) {
  pushMatrix();
  for (int i = 0; i < 4; i++) {
    rotateY(i * HALF_PI);
    pushMatrix();
    translate(0, 0, -.5 * boxSize);
    drawPlane(boxSize, boxSize, tex);
    popMatrix();
  }

  for (int i = 0; i < 2; i++) {
    rotateX(HALF_PI + i * HALF_PI);
    pushMatrix();
    translate(0, 0, -.5 * boxSize);
    drawPlane(boxSize, boxSize, tex);
    popMatrix();
  }
  popMatrix();
}

void drawPlane(float w, float h, PImage img) {
  float halfWidth = .5 * w;
  float halfHeight = .5 * h;
  beginShape();
  textureMode(NORMAL);
  texture(img);
  vertex(-halfWidth, -halfHeight, 0, 0, 0);
  vertex(-halfWidth, halfHeight, 0, 0, 1);
  vertex(halfWidth, halfHeight, 0, 1, 1);
  vertex(halfWidth, -halfHeight, 0, 1, 0);
  endShape(CLOSE);
}

<canvas width="200px" height="200px"></canvas>

body {
  background-color: #fff;
}

</style> <script type="text/javascript"> window.addEventListener('load', function() {
  var scripts=document.body.getElementsByTagName('script');
  var canvases=document.body.getElementsByTagName('canvas');
  new Processing(canvases[0], scripts[0].text);
}

, false);
// Here prevent javascript in body from throwing error
</script> <style>