The temperature dependence of enthalpy is determined by a parameter called the specific heat capacity (at constant pressure), Cp. If Cp is > 0, then enthalpy will increase with increasing temperature, whereas if it is < 0, enthalpy will decrease with increasing temperature. This is described by Kirchoff's law of thermodynamics. One thing to remember is that Cp itself might be temperature dependent (i.e. not necessarily a constant) in which case without knowing the functional dependence of Cp on T, it might be difficult to predict beforehand how H will change with T.
Andrew Lane has give the fullest answer to the question, for it's really important, wht happens to entropy in the same vein ? The heat capacity is really the key parameter to follow in this respect. This could in principle be done by micro-calorimetry, for example, in addition to building the conventional Arrhenius and/or van't Hoff plot for the reaction under study... Thus, you'll also see whether there is a valid entropy-enthalpy compensation. Otherwise, you could investigate a row of the pertinent homological reactions to find their thermodynamic/kinetic parameters - and how the latter depend on the temperature ...