I want to calculate the temperature and humidity via sense hat integrated with Raspberry Pi 3. Is there any free API for SMS available currently?
It's unlikely that you'll find a free API for sending text messages, simply because sending them isn't cost free.
That said, you're looking for an SMS Gateway. Many US mobile providers offer free gateways that will convert an email sent to a specific address into a text message to the corresponding telephone number. This may also be the case in other countries, although you will need to check with the providers you wish to communicate with.
Otherwise, you'll need to use a paid SMS Gateway, such as Twilio. Pricing varies by volume and location, but they do offer a free trial balance to test with, before you commit to paying per message. They offer libraries for various languages and the documentation is pretty extensive, which is what you'd expect if you did choose to pay for the service.
Other paid gateways likely have similar pricing structures and features, and you'll need to take a look around the market for the most appropriate for your use case.