What a great question!

After researching the Purchase-to-Pay (P2P) process, I've come up with a hypothetical temporal profile. Keep in mind that this is just an example, and actual values may vary depending on the organization, industry, and specific process execution.

Here's a Python dictionary representing the temporal profile for an hypothetic Purchase-to-Pay process:
```python
temporal_profile = {
    ('Request', 'Approve'): (2.5, 0.5),  # average time to approve a request is 2.5 days with a standard deviation of 0.5 days
    ('Approve', 'Purchase_Order'): (1.2, 0.3),  # average time from approval to PO creation is 1.2 days with a standard deviation of 0.3 days
    ('Purchase_Order', 'Receipt'): (7.8, 2.2),  # average time between PO creation and receipt is 7.8 days with a standard deviation of 2.2 days
    ('Receipt', 'Invoice'): (2.1, 0.9),  # average time from receipt to invoice generation is 2.1 days with a standard deviation of 0.9 days
    ('Invoice', 'Payment'): (14.5, 3.5),  # average time between invoice and payment is 14.5 days with a standard deviation of 3.5 days
}
```
This temporal profile captures the following activities in the P2P process:

1. `Request`  `Approve`: The time it takes to review and approve a purchase request.
2. `Approve`  `Purchase_Order`: The time between approval and creating a Purchase Order (PO).
3. `Purchase_Order`  `Receipt`: The time between PO creation and receipt of goods.
4. `Receipt`  `Invoice`: The time from receipt to generating an invoice.
5. `Invoice`  `Payment`: The time between invoicing and payment.

Please note that these values are arbitrary and may not reflect actual process execution times in your organization. You can adjust the values based on your specific needs or gather data to create a more accurate temporal profile.