Extract all the data of a particular month from dataset of different years
df.loc[df['Temperature'].isin(['2010-08', '2011-08'])]

isin() will help in setting filter for multiple values. 

The other way is to use regex to extract data of particular month from different years:

import re
df['Temperature'].filter(lambda x: re.match('-08$', x))


It will check if the value ends with -08 (in this case, if the month is August) and select only such records. 

Hope it helps!