  We present an analytic five-loop calculation for the additive renormalization
constant A(u,\epsilon) and the associated renormalization-group function B(u)
of the specific heat of the O(n) symmetric \phi^4 theory within the minimal
subtraction scheme. We show that this calculation does not require new
five-loop integrations but can be performed on the basis of the previous
five-loop calculation of the four-point vertex function combined with an
appropriate identification of symmetry factors of vacuum diagrams. We also
determine the amplitude function F+(u) of the specific heat in three dimensions
for n=1,2,3 above T_c and F-(u) for n=1 below T_c up to five-loop order,
without using the \epsilon=4-d expansion. Accurate results are obtained from
Borel resummations of B(u) for n=1,2,3 and of the amplitude functions for n=1.
Previous conjectures regarding the smallness of the resummed higher-order
contributions are confirmed. Combining our results for B(u) and F+(u) for
n=1,2,3 with those of a recent three-loop calculation of F-(u) for general n in
d=3 dimensions we calculate Borel resummed universal amplitude ratios A+/A- for
n=1,2,3. Our result for A+/A- = 1.056 +/- 0.004 for n=2 is significantly more
accurate than the previous result obtained from the \epsilon expansion up to
O(\epsilon^2) and agrees well with the high-precision experimental result A+/A-
= 1.054 +/- 0.001 for He(4) near the superfluid transition obtained from a
recent experiment in space.
