参考 stackoverflow 上 How do I add a custom claim to authentication cookie 问题的回答，通过下面的代码解决了

var authenticateResult = await HttpContext.AuthenticateAsync();
if (authenticateResult.Succeeded)
{
    var claimsIdentity = authenticateResult.Principal.Identity as ClaimsIdentity;
    if (claimsIdentity != null)
    {
        if (!claimsIdentity.HasClaim(c => c.Type == CnblogsClaimTypes.BlogId))
        {
            claimsIdentity.AddClaim(new Claim(CnblogsClaimTypes.BlogId, user.BlogId.ToString()));
            await HttpContext.SignInAsync(authenticateResult.Principal, authenticateResult.Properties);
        }
    }
}