You can add the time tracking component for custom objects in Salesforce.

To add the lighting component to custom objects you should first create a lookup field in TT Details object for the custom object.

Click on Setup, go to object manager and click on TT detail, click the new button from Fields and Relationship string and select Data Type as "Lookup Relationship" and select the custom object from the related to list and save it.

Once the lookup is created for TT details and Custom objects, you can add time tracking component for the custom object.