You can log time to other objects by adding a lookup field to the Time object and related layouts.

How do I log time against another object other than a matter?

To log time against an object other than Matters in your org, follow the steps below:

1. Create a lookup field to the desired object on the Time object.
2. Add a lookup field to the Time field set for visibility in our New/Edit Time form.
3. Add a lookup field to the Time Page Layout or Record Page.
4. Add a time-related list with the standard New button enabled on the desired object's page layout.

The caveat is that there are no background or flow automation actions when extending this to other objects outside of the Matter object that we have provided. Our Time form is a custom component-based form that cannot be customized beyond adding fields via fieldSets. If you wish to roll up the Time summary to the object, this will need to be implemented via a flow or Apex trigger.  If you need further instructions on adding a field to the Time object, follow this link.