activity_menu_app_bar_1.xml produces the desired effect, but the toolbar moves up: as per the height set in the empty CollapsingToolbarLayout before adding the elevation. activity_menu_app_bar_2.xml does not work, it adds elevation by default without scrolling.

activity_menu_app_bar_1.xml

<android.support.design.widget.AppBarLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:background="@android:color/transparent"
    android:id="@+id/activity_menu_app_bar">

    <android.support.design.widget.CollapsingToolbarLayout
        android:layout_width="match_parent"
        android:layout_height="1dp"
        android:minHeight="0.1dp"
        app:layout_scrollFlags="scroll|exitUntilCollapsed"/>

    <include layout="@layout/activity_menu_toolbar"/> 

</android.support.design.widget.AppBarLayout>

activity_menu_app_bar_2.xml

<android.support.design.widget.AppBarLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:background="@android:color/transparent"
    android:id="@+id/activity_menu_app_bar">

    <android.support.design.widget.CollapsingToolbarLayout
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:minHeight="1dp"
        app:layout_scrollFlags="scroll|exitUntilCollapsed">

        <include layout="@layout/activity_menu_toolbar"/> 
    
    </android.support.design.widget.CollapsingToolbarLayout>

</android.support.design.widget.AppBarLayout>

activity_menu_toolbar.xml

<RelativeLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:id="@+id/activity_menu_toolbar">

    <ImageButton
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:padding="@dimen/spacing_normal"
        android:background="?android:selectableItemBackgroundBorderless"
        android:src="@drawable/ic_search"
        android:layout_alignParentStart="true"
        android:layout_centerVertical="true"
        android:id="@+id/activity_menu_toolbar_search_button"/>

    <TextView
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="@string/app_name"
        android:textSize="@dimen/font_large"
        android:textStyle="bold"
        android:textAllCaps="true"
        android:textColor="@color/textLight"
        android:layout_centerInParent="true"
        android:id="@+id/activity_menu_toolbar_title_text"/>

    <ImageButton
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:padding="@dimen/spacing_normal"
        android:background="?android:selectableItemBackgroundBorderless"
        android:src="@drawable/ic_settings"
        android:layout_alignParentEnd="true"
        android:layout_centerVertical="true"
        android:id="@+id/activity_menu_toolbar_bookmark_button"/>

    <View
        xmlns:android="http://schemas.android.com/apk/res/android"
        android:layout_width="match_parent"
        android:layout_height="0.8dp"
        android:layout_alignParentBottom="true"
        android:background="@color/menuDivider"/>

</RelativeLayout>

try

<CoordinatorLayout>
  <AppBarLayout>
    <Toolbar />
    <android.support.v4.widget.Space
            android:layout_width="0dp"
            android:layout_height="1px"
            app:layout_scrollFlags="scroll|exitUntilCollapsed"
            android:minHeight="0px" />
  </AppBarLayout>
  ...
</CoordinatorLayout>

You'll get a 1px "collapse" but the Toolbar above won't move.

For some reason, it works only when I place the Space above the Toolbar and that again moves the Toolbar. Placing the Space below the Toolbar does not set any elevation at all. I tested it on a device and the emulator, both on API 25.

I enabled layout bounds and I noticed the Space wasn't collapsing, it stays at the bottom. I also tried wrapping the Toolbar and Space in a vertical linear layout and tried using a plain View instead of Space. But no luck :(

Say, why does AppBarLayout have a default stateListAnimator ? When does it ever get changed? Which states does it have, by default?