The last modified field is a great filter to use when wanting to see if the Invoices on the sale endpoint have been updated or edited from the API or UI however there are a few key areas to note that will not update this field.
The last modified field will only update with a direct action to the invoice, this means a payment to the invoice would not be included as a direct action.
A direct action is a field directly on the invoice being updated e.g a line item, date and Customer PO just to name a few.
Endpoints with the Last Modified Field available:
If you are encountering this situation you can use the field on invoices for Last Payment Date which can be used to filter e.g
?$filter=LastPaymentDate eq datetime'2020-12-21
If you have any questions, feel to log a ticket with the Customisation & Integration Team.