I connected everything like in the blogpost: An Arduino Library for the ADT7310 SPI Temperature Sensor – The inability to follow simple instructions
I got it to work by removing the sd-card. Even though I set digital pin 4 to high and cast it as an output. It still kept messing with my temperature values. But once I removed the card from the sd-card-slot it works just fine. I still was having some other problem with random values being read out, but I solved that problem by doing a NOP to the sensor after I used the ethernetshield.