t.rast.series is a simple wrapper for the raster module r.series. It supports a subset of the aggregation methods of r.series.
t.rast.series input=tempmean_monthly output=tempmean_average method=average
t.rast.series input=tempmean_daily output=tempmean_season method=average \ where="start_time >= '2012-06' and start_time <= '2012-08'"
t.rast.series input=tempmean_monthly \
    method=average output=tempmean_january \
    where="strftime('%m', start_time)='01'"
# equivalently, we can use 
t.rast.series input=tempmean_monthly \
    output=tempmean_january method=average \
    where="start_time = datetime(start_time, 'start of year', '0 month')"
# if we want also February and March averages
t.rast.series input=tempmean_monthly \
    output=tempmean_february method=average \
    where="start_time = datetime(start_time, 'start of year', '1 month')"
t.rast.series input=tempmean_monthly \
    output=tempmean_march method=average \
    where="start_time = datetime(start_time, 'start of year', '2 month')"
for i in `seq -w 1 12` ; do 
  for m in average stddev minimum maximum ; do 
    t.rast.series input=tempmean_monthly method=${m} output=tempmean_${m}_${i} \
    where="strftime('%m', start_time)='${i}'"
  done
done