Relevant code

Form1

From designer code

trackBar1.Value = 100;
BackColor = Color.Gray;

From code-behind

private void trackBar1_Scroll(object sender, EventArgs e)
{
    userControl11.SetZoomFactor(trackBar1.Value / 100F);
}

UserControl1

internal float MyBaseWidth;

public UserControl1()
{
    InitializeComponent();

    MyBaseWidth = Width;

    SetZoomFactor(1);
}

protected override void OnPaint(PaintEventArgs e)
{
    base.OnPaint(e);

    e.Graphics.SmoothingMode = SmoothingMode.HighQuality;
    e.Graphics.CompositingQuality = CompositingQuality.HighQuality;
    e.Graphics.InterpolationMode = InterpolationMode.HighQualityBicubic;

    Pen p = new Pen(Color.Yellow);
    e.Graphics.DrawPath(p, GraphicsPathWithBorder);
}

internal GraphicsPath GraphicsPathWithBorder;

internal void SetZoomFactor(float z)
{
    Width = (int)(MyBaseWidth * z);

    GraphicsPathWithBorder = RoundedCornerRectangle(ClientRectangle);
    Region = new Region(GraphicsPathWithBorder);
}

internal static GraphicsPath RoundedCornerRectangle(Rectangle r)
{
    GraphicsPath path = new GraphicsPath();
    float size = 10 * 2F;

    path.StartFigure();

    path.AddArc(r.X, r.Y,
        size, size, 180, 90);
    path.AddArc((r.X + (r.Width - size)), r.Y,
        size, size, 270, 90);
    path.AddArc((r.X + (r.Width - size)), (r.Y + (r.Height - size)),
        size, size, 0, 90);
    path.AddArc(r.X, (r.Y + (r.Height - size)),
        size, size, 90, 90);

    path.CloseFigure();

    return path;
}

Initial screenshot

Screenshot after using the trackbar

The right side of the yellow border becomes invisible after zooming out, and when zooming in there are multiple yellow borders on the right side.

Update:

The answer Works, but there is a part of the control that goes beyond the border. Screenshot for top-right corner, for curveSize = 20:

and for curveSize = 24:

I suggest a slightly different method to draw the Border and the content of the User Control that should also cure the artifacts generated when the control is redrawn.

When you create a Region for a Control and then you paint the Region as it is, the outer borders of the painting are not anti-aliased: the aliased pixels fall outside the Region. The same effect of course is applies when a border is painted around the bounds of the Region.

Here, I apply a Scale Matrix and a Translate Matrix that scale and move the bounds of the Region on the inside of the outer Region that defines the control's bounds.
The size of the scale and the translate transformations are determined by the Pen size.
More informations on the Matrix usage here: Flip the GraphicsPath

In this case, when the borders are painted the outer, anti-aliased section of the border is inside the Region bounds and the anti-aliasing is preserved.
The background color of the Control is set to Color.Transparent (a User Control supports color transparency on its own).

I've also added a couple of (non decorated) properties that allow to define the inner Color (the Control's BackgroundColor) and Size and Color of the Border. The rest is more or less what it was before.

Sample results:

using System.Drawing;
using System.Drawing.Drawing2D;

public partial class RoundControl : UserControl
{
    private GraphicsPath GraphicsPathWithBorder;
    private float MyBaseWidth;
    private float m_PenSize = 2f;
    private Color m_BorderColor = Color.Yellow;
    private Color m_FillColor = Color.Green;

    public RoundControl()
    {
        this.ResizeRedraw = true;
        InitializeComponent();
        MyBaseWidth = Width;
    }

    public float BorderSize
    {
        get => this.m_PenSize;
        set {
            this.m_PenSize = value;
            this.Invalidate();
        }
    }

    public Color BorderColor
    {
        get => this.m_BorderColor;
        set {
            this.m_BorderColor = value;
            this.Invalidate();
        }
    }

    public Color FillColor
    {
        get => this.m_FillColor;
        set {
            this.m_FillColor = value;
            this.Invalidate();
        }
    }

    protected override void OnLayout(LayoutEventArgs e) {
        this.UpdateRegion();
        base.OnLayout(e);
    }

    protected override void OnPaint(PaintEventArgs e)
    {
        base.OnPaint(e);

        e.Graphics.SmoothingMode = SmoothingMode.AntiAlias;
        RectangleF rect = GraphicsPathWithBorder.GetBounds();
        float scaleX = 1 - ((m_PenSize + 1) / rect.Width);
        float scaleY = 1 - ((m_PenSize + 1) / rect.Height);
        using (Pen pen = new Pen(m_BorderColor, m_PenSize))
        using (Brush brush = new SolidBrush(m_FillColor))
        using (Matrix mx = new Matrix(scaleX, 0, 0, scaleY, pen.Width / 2, pen.Width / 2))
        {
            e.Graphics.Transform = mx;
            e.Graphics.FillPath(brush, GraphicsPathWithBorder);
            e.Graphics.DrawPath(pen, GraphicsPathWithBorder);
        }
    }

    private void UpdateRegion() {
        GraphicsPathWithBorder = RoundedCornerRectangle(ClientRectangle);
        Region = new Region(GraphicsPathWithBorder);
        this.Invalidate();
    }

    internal void SetZoomFactor(float z) {
        int newWidth = (int)(MyBaseWidth * z);
        if (newWidth <= (30 + this.m_PenSize * 2)) return;
        this.Width = newWidth;
        this.UpdateRegion();
    }

    private GraphicsPath RoundedCornerRectangle(Rectangle r)
    {
        GraphicsPath path = new GraphicsPath();
        float curveSize = 10 * 2.4F;

        path.StartFigure();
        path.AddArc(r.X, r.Y, curveSize, curveSize, 180, 90);
        path.AddArc(r.Right - curveSize, r.Y, curveSize, curveSize, 270, 90);
        path.AddArc(r.Right - curveSize, r.Bottom - curveSize, curveSize, curveSize, 0, 90);
        path.AddArc(r.X, r.Bottom - curveSize, curveSize, curveSize, 90, 90);
        path.CloseFigure();
        return path;
    }
}