You can easily add a second legend by adding the line:
ax2.legend(loc=0)
You'll get this:
But if you want all labels on one legend then you should do something like this:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc
rc('mathtext', default='regular')
time = np.arange(10)
temp = np.random.random(10)*30
Swdown = np.random.random(10)*100-10
Rn = np.random.random(10)*100-10
fig = plt.figure()
ax = fig.add_subplot(111)
lns1 = ax.plot(time, Swdown, '-', label = 'Swdown')
lns2 = ax.plot(time, Rn, '-', label = 'Rn')
ax2 = ax.twinx()
lns3 = ax2.plot(time, temp, '-r', label = 'temp')
# added these three lines
lns = lns1+lns2+lns3
labs = [l.get_label() for l in lns]
ax.legend(lns, labs, loc=0)
ax.grid()
ax.set_xlabel("Time (h)")
ax.set_ylabel(r"Radiation ($MJ\,m^{-2}\,d^{-1}$)")
ax2.set_ylabel(r"Temperature ($^\circ$C)")
ax2.set_ylim(0, 35)
ax.set_ylim(-20,100)
plt.show()
Which will give you this:
