Here's a hypothetical temporal profile for a Purchase-to-Pay (P2P) process, assuming the following activities: 'Request', 'Approve', 'Purchase', 'Receive', 'Invoice', 'Pay'. I've made some assumptions about the average times and standard deviations based on common business processes.

```python
p2p_temporal_profile = {
    ('Request', 'Approve'): (14400, 7200),  # Avg: 4 hours, Std Dev: 2 hours
    ('Approve', 'Purchase'): (86400, 25200),  # Avg: 1 day, Std Dev: 7 hours
    ('Purchase', 'Receive'): (259200, 72000),  # Avg: 3 days, Std Dev: 20 hours
    ('Receive', 'Invoice'): (43200, 10800),  # Avg: 12 hours, Std Dev: 3 hours
    ('Invoice', 'Pay'): (172800, 43200),  # Avg: 2 days, Std Dev: 12 hours
    ('Request', 'Purchase'): (259200, 72000),  # Avg: 3 days, Std Dev: 20 hours
    ('Request', 'Receive'): (362880, 93600),  # Avg: 4.2 days, Std Dev: 26 hours
    ('Approve', 'Receive'): (259200, 72000),  # Avg: 3 days, Std Dev: 20 hours
    ('Approve', 'Pay'): (432000, 108000),  # Avg: 5.1 days, Std Dev: 29.9 hours
    ('Purchase', 'Pay'): (362880, 93600),  # Avg: 4.2 days, Std Dev: 26 hours
}
```

In this profile:

- Times are given in seconds.
- Activities like 'Approve' and 'Pay' have longer standard deviations to account for varying approval times and payment cycles.
- Indirect activity pairs (e.g., 'Request', 'Pay') have longer average times and standard deviations, as they span the entire P2P process.
- The 'Receive' activity has a shorter average time and standard deviation compared to other activities, as receiving goods is typically a quicker process compared to approvals and payments.