@Alejandro Miranda
Thank you for reaching out. I see you have mentioned that a screenshot was attached in the question but it seems to be missing. Can you please share the screenshot again and also please share a screenshot of the relevant Azure Font Door rule set-up if any for this scenario?
My question is... when you use Azure Front Door the customized domain appears too when you use Google as Identity Provider, in the consent screen? (I attached a picture)
This behavior will depend upon how the rule is set:
If it is set to redirect then the customized domain will not appear.
If the rule is not set to redirect, then the backend will authenticate with the Google Auth and it will not appear as identity provider to the user.
Regarding your 2nd question above
I could see that Azure Front Door has a fixed cost of $35, is correct?
As per the pricing document here
35$ is the monthly base fees (Billed hourly and only for number of hours used) Front Door Standard SKU. There will be additional charge for Outbound Data Transfer from Edge to the Client, Outbound Data Transfer from Edge to the Origin and Request Pricing.
Thank you!