How to update user claims stored in authentication cookie in .NET Core 6

I have a Razor Pages app developed using .NET Core 6. The app works as a client and connects to an API. The API has JWT Access Token/Refresh Token authentication. The login endpoint of the API returns access token and refresh token.Using cookie authentication I store the tokens as claim in authentication cookie.

var claims = new List<Claim>
        {
            new Claim(ClaimTypes.Name, Login.Email),
            new Claim("Token", loginResponse.Token),
            new Claim("RefreshToken", loginResponse.RefreshToken)
        };

I then get the tokens using extension methods

public static class CommonExtensions
{
    public static string? GetToken(this HttpContext context)
    {
        return context.User.Claims.Single(x => x.Type == "Token").Value.ToString();
    }

    public static string? GetRefreshToken(this HttpContext context)
    {
        return context.User.Claims.Single(x => x.Type == "RefreshToken").Value.ToString();
    }
}

When my access token expires I refresh it, remove existing claims and add new ones with the updated token.

var identity = User.Identity as ClaimsIdentity;
identity.RemoveClaim(identity.FindFirst("Token"));
identity.AddClaim(new Claim("Token", response.Token));

identity.RemoveClaim(identity.FindFirst("RefreshToken"));
identity.AddClaim(new Claim("RefreshToken", response.RefreshToken));

However the subsequent requests keep using the expired token. What is the way to update the claims correctly?