It’s a lovely common word, 164 uses in The Hobbit, and used absolutely evenly in Chapter 8. It’s in the war chapters that the word is used fewer times, and I suspect that it’s still used proportionally to word count. When I have a moment to think a thought all the way through, this one might be well worth graphing to find a baseline, evenly distributed word.

• 1.004 which they have twice a day when they can get it).
• 1.005 and there they remained to the end of their days.
• 1.010 we have all the day before us!’
• 1.024 The next day
• 1.073 We must away ere break of day
• 1.077 We must away, ere break of day,
• 1.082 We must away, ere break of day,
• 1.091 before the break of day start on our long journey,
• 1.122 in those days.
• 1.122 Altogether those were good days for us,
• 1.123 in the North in those days,
• 1.123 One day he flew up into the air
• 1.123 in those days,
• 1.123 and it saved my life that day) –
• 1.123 in those days.
• 1.124 and said that one day
• 1.144 We must away, ere break of day,
• 1.145 It was long after the break of day,
• 2.019 To the end of his days
• 2.027 as they rode forward all day,
• 2.028 for the weather that day had taken a nasty turn.
• 2.029 and had been all day;
• 2.106 And there they stand to this day,
• 2.115 in these parts and days;
• 2.125 in a few days now,
• 3.001 They did not sing or tell stories that day,
• 3.001 nor the next day,
• 3.001 nor the day after.
• 3.001 Already they seemed only a day’s easy journey from the feet of the nearest.
• 3.010 when the day began to fail.
• 3.026 when sun has been all day on the snow far up above.
• 3.031 and days that are good to spend
• 3.031 fourteen days at least,
• 3.032 In those days of our tale
• 3.034 in a few days there.
• 3.035 That day he looked at the swords they had brought
• 3.042 as the day when they were written.
• 3.044 with the last light of Durin’s Day
• 3.046 “Then what is Durin’s Day?”
• 3.047 “The first day of the dwarves’ New Year,”
• 3.047 the first day of the last moon of Autumn
• 3.047 We still call it Durin’s Day
• 3.047 in these days to guess when such a time will come again.”
• 4.002 Long days after they had climbed
• 4.003 “and perhaps it will be Durin’s Day”
• 4.004 until one day they met a thunderstorm
• 4.012 – as he did that day
• 4.025 but in those days
• 5.009 and over it to the days beyond.
• 5.037 Also they reminded him of days when he had been less lonely
• 5.085 in the endless dark days.
• 5.087 in the old days
• 5.128 a glimpse of endless unmarked days
• 6.061 But in those days they sometimes used to go on raids,
• 6.062 or in the bright day.
• 6.062 there would have been none left there next day;
• 6.088 of the last three days
• 7.009 in after days the King of All Birds
• 7.012 I can give you a day or two more.
• 7.023 The day will come when they will perish
• 7.044 if it won’t take all day,’
• 7.082 in the old days he had known the very part of the mountains
• 7.094 and South a day’s ride before them,
• 7.098 there shadows lay by night and day,
• 7.113 But he saw no sign of the wizard all that day until the evening.
• 7.113 as they had been all day.
• 7.114 and where have you been all day yourself?’
• 7.130 for at a place a few days’ ride due north of the Carrock
• 7.133 So they rode now for two more days,
• 7.134 Next day they started before dawn,
• 7.151 and with a tremendous slice of luck you may come out one day
• 8.004 in but not to spend summer days in,
• 8.007 as days followed days,
• 8.007 as days followed days,
• 8.008 when one day they found their path blocked by a running water.
• 8.035 in the following days.
• 8.036 in the last few days,
• 8.036 In a few days a time came when
• 8.037 About four days from the enchanted stream
• 8.039 Two days later
• 8.050 This they did all that day,
• 8.074 till day came with some little light,
• 8.107 and the next day
• 8.110 and then we’ll hang him head downwards for a day or two.’
• 8.130 and all the sounds of the battle next day,
• 8.131 even in those days they were wary.
• 8.133 In ancient days they had had wars with some of the dwarves,
• 9.001 The day after the battle with the spiders
• 9.001 Such day as there ever was
• 9.012 day after day,’
• 9.012 day after day,’
• 9.013 What was his surprise one day
• 9.018 One day,
• 9.035 There would be a different expression on the face of the chief guard next day,
• 9.053 Seek the sunlight and the day,
• 9.053 Under sunlight, under day!
• 10.001 The day grew lighter
• 10.003 Those lands had changed much since the days
• 10.003 days which most people now remembered
• 10.005 late in the day the shores grew rocky,
• 10.007 but in the great days of old,
• 10.008 and Thrain would come back one day
• 10.018 in their young days.
• 10.037 and sang songs all day,
• 10.039 grew stronger every day.
• 10.039 For three days he sneezed
• 10.045 So one day,
• 11.001 In two days going
• 11.001 At the end of the third day,
• 11.003 The next day they set out again.
• 11.007 in the days
• 11.007 on the day the Dragon came.
• 11.013 shadowed all day by cliff
• 11.013 day by day they toiled
• 11.013 day by day they toiled
• 11.013 Day by day they came back to their camp without success.
• 11.013 Day by day they came back to their camp without success.
• 11.014 Fili and Kili and the hobbit went back one day
• 11.024 said Thorin one day.
• 11.028 Next day the dwarves all went wandering
• 11.028 All day Bilbo sat gloomily
• 12.004 in the old days’ –
• 12.015 in the days when all the world was wonderful.
• 12.039 By day we can no doubt creep out
• 12.067 that in his young days had been called Esgaroth.
• 13.001 At last after days
• 13.001 and days of waiting,
• 13.048 before them stood the broad light of day.
• 13.057 and the day between had gone by
• 13.067 in the old days we used always to keep watchmen,
• 13.067 in the days of our prosperity,
• 13.071 in all the days of Smaug’s dominion.
• 14.001 two days before.
• 14.012 enough to cover all the land with a mist for days;
• 14.038 and in the days that followed
• 14.039 although it was then only the third day after the fall of Smaug.
• 14.042 in these days
• 14.042 Only five days after the death of the dragon
• 14.044 in eleven days
• 15.026 you will see that the dwarves still had some days before them.
• 15.028 They were four days gone,
• 15.033 That day the camp was moved to the east of the river,
• 15.033 as they had not done for many a day.
• 15.049 in the past days
• 16.001 Now the days passed slowly
• 16.003 for as the weariness of the days grew heavier,
• 16.004 were now within about two days’ march of Dale,
• 16.031 is now less than two days’ march off,
• 16.044 For the first time for many a day
• 17.001 Next day the trumpets rang early
• 17.031 That day passed
• 17.031 The next day the wind shifted west,
• 17.054 Day drew on.
• 18.001 A cloudless day,
• 18.024 perished on that day,
• 18.030 Actually it was some days
• 18.042 for many a long day.
• 18.051 in those days beside goblins;
• 18.051 In their day the last goblins were hunted
• 19.028 a day came at last when they were
• 19.037 it was a great deal more than a nine days’ wonder.
• 19.039 in the quiet days before the Unexpected Party.
• 19.040 he remained very happy to the end of his days,
• 19.045 in his day the rivers run with gold.’