Final answer:
The approval for all officer Temporary Limited Duty (TLD) and third period or higher enlisted TLD in the military is typically managed by higher command authorities, which can vary by the branch of service.
Step-by-step explanation:
The authority that approves all officer Temporary Limited Duty (TLD) and third period or higher enlisted TLD within the military is often specific to the branch of service and can vary by country and their respective military regulations.
In general, such approvals are given by higher command authority, such as personnel or medical departments within the service. These could be entities like the Bureau of Medicine and Surgery (BUMED) in the case of the US Navy, or similar directorates in other branches.
High-level endorsements for extended or repeat TLD requests ensure that members of the armed forces receive the necessary care and rehabilitation required while also managing the staffing and readiness concerns of the military unit.
The exact approving authority will always follow the strict guidelines and procedures established by the military branch in question.