Section 262 of the Law of Obligations grants you that right. It is not automatic: you need the landlord’s written consent before any temporary tenant moves in.
The landlord has between 14 and 30 days to reply. If they do not reply, it is assumed that they have tacitly agreed. However, a written reply is always safer.
A refusal is only permissible on three grounds: if you fail to inform the tenant of the terms, if the terms are unfair, or if the subletting would cause serious disruption. Any other refusal has no legal basis and may be appealed to the cantonal tenancy conciliation authority.
As for the rent: charging the same amount as you pay is fair. For a furnished flat, a mark-up of up to 10–20% is generally accepted. Anything more than that may be considered exploitative by the landlord. And if the sublet lasts for more than two consecutive years, the landlord may argue that the flat is no longer your main residence. That is a real risk.