The answer would be no, in general.
For general substances/solids the heat capacity is not constant with temperature. For an ideal gas, the heat capacity is a constant because its (internal) energy is proportional to the (absolute) temperature, i.e. any increase in temperature is proportional to an increase in energy.
The heat capacity of solids is rather more complicated --- for example, in metals there are contributions from the conduction electrons and contributions from the collective (that is,. coupled) oscillations of the ion cores, the "phonons", and even each of them separately are not constants
So, if by background you mean the heat capacity of a "heat reservoir" or a calorimeter, or whatever holds your sample, its non constancy in temperature has to be taken into account. The particular details of the measurement have to be taken into account of course, for example above a certain temperature the heat capacity of the background may be "almost" constant
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