Sure, for a hypothetical Purchase-to-Pay (P2P) process, let's first outline the typical steps in such a process and then provide an example of a temporal profile. Note that the times between activities can vary significantly between organizations depending on their specific processes, policies, and systems used. Here's a simplified version:

1. **Create Purchase Request (PR):** The process starts with the creation of a purchase request.
2. **Approve Purchase Request:** The purchase request is then reviewed and approved.
3. **Create Purchase Order (PO):** Following approval, a purchase order is created and sent to the supplier.
4. **Receive Goods/Services:** Once the goods or services are received, a receipt is created.
5. **Receive Invoice:** The invoice is received from the supplier.
6. **Approve and Pay Invoice:** Finally, the invoice is approved and payment is processed.

Given these steps, we can build a hypothetical temporal profile based on a general understanding of how long each step might take. It's important to understand that these are purely illustrative durations:

- **Create Purchase Request (PR) to Approve Purchase Request:** On average, this might take 1 day (86,400 seconds) with a standard deviation of 2 hours (7,200 seconds).
- **Approve Purchase Request to Create Purchase Order (PO):** This could take roughly 2 days on average (172,800 seconds) with a standard deviation of 4 hours (14,400 seconds), reflecting various approval speeds.
- **Create Purchase Order (PO) to Receive Goods/Services:** The lead time here can be significant, let's assume an average of 14 days (1,209,600 seconds) with a variance (standard deviation) of 2 days (172,800 seconds), considering different delivery schedules and possible delays.
- **Receive Goods/Services to Receive Invoice:** This could happen relatively quickly, with an average of 2 days (172,800 seconds) but a smaller standard deviation of 1 day (86,400 seconds) due to processing times.
- **Receive Invoice to Approve and Pay Invoice:** This process might take an average of 5 days (432,000 seconds), with a standard deviation of 1 day (86,400 seconds) considering internal approval and processing variability.

Here's how we might express this temporal profile in a Python dictionary:

```python
temporal_profile = {
    ('Create PR', 'Approve PR'): (86400, 7200), 
    ('Approve PR', 'Create PO'): (172800, 14400), 
    ('Create PO', 'Receive Goods/Services'): (1209600, 172800),
    ('Receive Goods/Services', 'Receive Invoice'): (172800, 86400),
    ('Receive Invoice', 'Approve and Pay Invoice'): (432000, 86400)
}
```

This dictionary represents the hypothetical average times and standard deviations (in seconds) between key steps in a Purchase-to-Pay process. Remember, the actual values could widely vary based on organizational procedures, efficiency, and the specific nature of purchases (types of goods or services, supplier relationships, etc.).