Specific Heat Capacity of Heavy Water
I posted an answer here the other day which was completely wrong! I apologise and have left the incorrect text below for posterity to record how wrong people who should know better can be!

Looking on the NIST fluid properties database I see that the specific heat capacity (i.e. the heat capacity per unit mass) of D2O is just 1.5% higher than the specific heat capacity of H2O. So at 298 K this makes the molar heat capacity of D2O (84.963 J/K/mol) an astonishing 12.7% higher than H2O (75.38 J/K/mol). I am astounded and have no explanation. 

Everything below this line is WRONG!

The molar heat capacity is almost exactly the same. i.e. it takes the same energy to heat the same number of molecules by one degree Celsius. 
The difference in the *specific* heat is just because of the difference in density. D2O molecules have a relative molar mass of about 20 compared with 18 for H2O. Since the bond lengths are similar, the density is higher by a factor 20/18 - about 11% higher. So a given mass of D2O has about 11% fewer molecules in it.