You need to give rosemary special conditions to keep it happy indoors. Source: http://www.thekitchn.com
Rosemary (Rosmarinus officinalis) is a popular cooking herb often grown in home gardens. It’s a half-hardy shrub and will grow outdoors year round in zones USDA 8 to 10, sometimes even in zone 7 (the cultivar ‘Arp’ is said to be hardy—sometimes!—in zone 6). Still, gardeners in cold climates like mine have to bring it indoors for the winter … and often find it difficult to maintain.
That’s because rosemary comes from a Mediterranean climate, where summers are hot and dry and winters cool and rainy. It suffers greatly from the heat we maintain in our homes in winter. The more your conditions are like those of its place of origin, though, the better it will perform. Try moving it to that rarest of indoor locations: a cool yet brightly lit room. Night temperatures from about 38 to 50˚ F (3 to 10˚ C) are best, but 60˚ F (15˚ C) is acceptable …and it can take a few degrees of frost in a pinch. If you can’t give it full sun in this frigid spot, consider adding artificial light.
Overwatering can be a problem in such a cool spot. The plant will use less water than normal and evaporation will be reduced, so you’ll probably find you’ll only need to water every 2 to 3 weeks. That does not mean to let it dry out completely, though! Continue “standard watering”: a good soaking when the soil is dry to the touch. It’s just that “dry to the touch” will occur less frequently than in a heated room.