New high-resolution dissociative recombination rate coefficients of rotationally cool and hot H3+ in the vibrational ground state have been measured with a 22-pole trap setup and a Penning ion source, respectively, at the ion storage ring TSR. The experimental results are compared with theoretical calculations to explore the dependence of the rate coefficient on ion temperature and to study the contributions of different symmetries to probe the rich predicted resonance spectrum. The break-up energy was investigated by fragment imaging to derive internal temperatures of the stored parent ions under differing experimental conditions. A systematic experimental assessment of heating effects is performed which, together with a survey of other recent storage-ring data, suggests that the present rotationally cool rate-coefficient measurement was performed at 380^{+50}_{-130} K and that this is the lowest rotational temperature so far realized in storage-ring rate-coefficient measurements on H3+. This partially supports the theoretical suggestion that higher temperatures than assumed in earlier experiments are the main cause for the large gap between the experimental and theoretical rate coefficients. For the rotationally hot rate-coefficient measurement a temperature of below 3250K is derived. From these higher-temperature results it is found that increasing the rotational ion temperature in the calculations cannot fully close the gap between the theoretical and experimental rate coefficients. Comment: 12 pages, 7 figures (11 subfigures), 3 tables