Is it permissible for an employee to take the cost of the last night of a business trip after having canceled his hotel reservation and stayed with a friend, knowing that the company covers all travel expenses?

If the company pays you amounts for your travel to perform tasks, regardless of the expenses, then these amounts are yours, and whatever remains of them belongs to you. However, if the company covers the actual expenses, you must return any remaining money, or inform them. If they permit you to keep it, then there is no harm.